\documentclass[12pt,draft]{amsart}
%%\usepackage{truetype}
\usepackage{amsmath,amsthm,latexsym,amscd,amsbsy,amssymb}
\setlength{\textwidth}{5.5in}

\font\fivrm=cmr5 \relax
%%\input{prepictex}
%%\input{pictex}
%%\input{postpictex}

%%%%% VERSION OF FEBRUARY 21, 1997 %%%%%%%%%%%

%%%%%%%%%%%%%%% BEGIN AUTHORS'' MACROS %%%%%%%%%%%%

\chardef\bslash=`\\ % p. 424, TeXbook
\newcommand{\ntt}{\series m\shape n\tt}
\newcommand{\cs}[1]{{\protect\ntt\bslash#1}}
\newcommand{\opt}[1]{{\protect\ntt#1}}
\newcommand{\env}[1]{{\protect\ntt#1}}
\makeatletter
\def\verbatim{\interlinepenalty\@M \@verbatim
  \leftskip\@totalleftmargin\advance\leftskip2pc
  \frenchspacing\@vobeyspaces \@xverbatim}
\makeatother
\hfuzz1pc

\makeatletter
  \def\dgt@k{\dg@DX=-3 \dg@DY=2 \dg@SIZE=3}
\makeatother

\makeatletter
  \def\dgt@kk{\dg@DX=3 \dg@DY=-1 \dg@SIZE=3}%
\makeatother

\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{pro}[thm]{Proposition}

\newtheorem*{A}{Theorem A}
\newtheorem*{B}{Theorem B}
\newtheorem*{C}{Theorem C}
\newtheorem*{D}{Theorem D}
\newtheorem*{E}{Theorem E}

\newtheorem*{F}{Theorem F}
\newtheorem*{G}{Theorem G}

\theoremstyle{definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{defin}[thm]{Definition}

\newcommand{\bb}{\bigstar_{\mathbb C}}

\numberwithin{equation}{section}

\def\hats #1{\widehat{\widehat{\hbox{$#1$}}}}
\font\f=msbm10
\font\eightrm=cmr8
%%%%%%%%%%% END AUTHORS' MACROS %%%%%%%%%%%%%
%%%%%%%%%%% BEGIN TEXT %%%%%%%%%%%%%%%%%%%%%%

\begin{document}

%%%%%%% Begin Topmatter %%%%%%%%%%

\title[Topological $\text{AE}(0)$-groups]{Topological $\text{AE}(0)$-groups}
\author{Alex Chigogidze}
\address{Department of Mathematics and Statistics,
University of Saskatche\-wan,
McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6,
Canada}
\email{chigogid@math.usask.ca}
\thanks{Author was partially supported by NSERC research grant.}

\keywords{Topological $\text{AE}(0)$-group, Polish group,
inverse spectrum}
\subjclass{Primary: 22A05; Secondary: 22F05}

%%%%%%% End topmatter %%%%%%%%%

\begin{abstract}{We investigate topological
$\text{AE}(0)$-groups class of which
contains the class of Polish groups as well as the class of
all locally compact groups. We establish the existence
of an universal $\text{AE}(0)$-group of a given weight
as well as the existence of an universal action of
$\text{AE}(0)$-group of a given weight on a $\text{AE}(0)$-space
of the same weight. A complete characterization of closed
subgroups of powers of the symmetric group $S_{\infty}$ is obtained.
It is also shown that every $\text{AE}(0)$-group is Baire
isomorphic to the product of Polish groups. These results
are obtained by using the spectral descriptions of
$\text{AE}(0)$-groups which are presented in Section \ref{S:spectral}.}
\end{abstract}

\maketitle
\markboth{A.~Chigogidze}{Topological $\text{AE}(0)$-groups}

\section{Introduction}\label{S:intro}

One of the main structure theorems for compact groups
(see \cite[ Chapters 6 \& 9]{hofmor98}) can be formulated as follows.

\begin{A}[\cite{hofmor98}, Theorem 9.24(ii)]\label{T:A}
Let $G$ be a connected compact group. Then there exists
a continuous homomorphism
\[ p \colon Z_{0}(G) \times \prod\{ L_{t} \colon t \in T\} \to G ,\]
where $Z_{0}(G)$ stands for the identity component of the
center of $G$, and $L_{t}$ is a simple, connected and simply
connected compact Lie group, $t \in T$, such that $\ker (p)$
is a zero-dimensional central subgroup of
$\displaystyle Z_{0}(G) \times \prod\{ L_{t} \colon t \in T\}$.
\end{A}

%%Note here that if $G$ (in Theorem A) itself is a connected
%%$n$-dimen\-si\-o\-nal compact Lie group,
%%then $\ker (p)$ is finite, the indexing set $T$ is also finite
%%and $Z_{0}(G)$ is a torus of codimension $|T|$ (i.e. $Z_{0}(G)$
%%is isomorphic to the product of $n-|T|$ copies of the
%%circle group $\text{\f T}$).

The above statement clearly shows that the classes of
zero-dimensional groups,
abelian groups and simple, simply connected Lie groups
play a central role in the
general theory of compact groups. Recently these classes of
topological groups
have been studied from the point of view
of absolute extensors in dimension $n$ (see \cite{chibook96}
for a comprehensive introduction into the theory of absolute
extensors in dimension $n$). The following three statements
show that such an approach is quite effective.

\begin{B}[\cite{bellchi93}]\label{T:B}
The following conditions are equivalent for a
zero-dimen\-si\-o\-nal topological group $G$:
\begin{itemize}
\item[(a)]
$G$ is topologically equivalent to the product
$(\text{\f Z}_{2})^{\tau} \times \text{\f Z}^{\kappa}$.
\item[(b)]
$G$ is an $\text{AE}(0)$-space.
\end{itemize}
\end{B}

\begin{C}[\cite{chi993}, Theorem E]\label{T:C}
The following conditions are equivalent for a compact
abelian group $G$:
\begin{itemize}
\item[(a)]
$G$ is a torus group (both in topological and algebraic senses).
\item[(b)]
$G$ is an $\text{AE}(1)$-compactum.
\end{itemize}
\end{C}

\begin{D}[\cite{chi972}, Corollary 1]\label{T:D}
The following conditions are equivalent for a non-trivial
compact group $G$:
\begin{itemize}
\item[(a)]
$G$ is a simple, connected and simply connected Lie group.
\item[(b)]
$G$ is an $\text{AE}(2)$-group with $\pi_{3}(G) = \text{\f Z}$.
\end{itemize}
\end{D}

The full classification problem for
non-abelian (see Theorem C) and for non-simply connected
compact $\text{AE}(1)$-groups remains open. On the other hand,
the following two statements provide a complete classification of
simply connected compact $\text{AE}(1)$-groups.

\begin{E}[\cite{chi972}, Theorem C]\label{T:E}
The following conditions are equivalent for a compact group $G$:
\begin{itemize}
\item[(a)]
$G$ is a simply connected $\text{AE}(1)$-compactum.
\item[(b)]
$G$ is an $\text{AE}(2)$-compactum.
\item[(c)]
$G$ is an $\text{AE}(3)$-compactum.
\item[(d)]
$G$ is a product of simple, connected and simply connected
compact Lie groups.
\end{itemize}
\end{E}

\begin{F}[\cite{chi972}, Corollary 2]\label{T:F}
There is no non-trivial compact
$\text{AE}(4)$-group.
\end{F}

We complete this brief survey by pointing out that, as shows the
following statement,
for locally compact
groups the restriction of being $\text{AE}(0)$-group is purely formal.

\begin{G}[Pontryagin-Haydon, \cite{pont46},
\cite{hay74}; \cite{chibook96}]\label{T:G}
Every locally compact group is an $\text{AE}(0)$-space.
\end{G}

Below we study $\text{AE}(0)$-groups.
The class of $\text{AE}(0)$-groups contains the class of all
(generally speaking, non-metrizable)
locally compact groups (Theorem G) as well as the class of
all Polish groups. Actually the class of Polish groups coincides
with the class of metrizable $\text{AE}(0)$-groups and forms a
foundation of the entire theory of $\text{AE}(0)$-groups. We
hope that results presented in Section \ref{S:spectral} do
indicate a potential for a non-trivial
theory of $\text{AE}(0)$-groups which unifies and generalizes
theories of locally compact and Polish groups (thus offering a
possible approach to the
corresponding question posed in \cite{pest99}).

In Section \ref{S:spectral} we present a spectral characterization of
$\text{AE}(0)$-groups in terms of well ordered continuous
inverse spectra (Theorem \ref{T:ae0}). This characterization
states that a non-metrizable topological group $G$ of weight $\tau$
is a $\text{AE}(0)$-group if and
only if it is the limit of a well ordered continuous inverse system
${\mathcal S}_{G} = \{ G_{\alpha}, p_{\alpha}^{\alpha +1},
\alpha < \tau \}$ of length $\tau$, consisting of
$\text{AE}(0)$-groups $G_{\alpha}$ and $0$-soft
limit homomorphisms
$p_{\alpha}^{\alpha +1} \colon G_{\alpha +1} \to G_{\alpha}$,
$\alpha < \tau$, so that $G_{0}$ is a Polish group and each homomorphism
$p_{\alpha}^{\alpha +1}$, $\alpha < \tau$, has a Polish kernel.

Obviously this result can not be accepted as the one providing a
satisfactory reduction of the non-metrizable case to the Polish
one. Of course, everything is fine if the weight of $G$ is
$\omega_{1}$ -- in such a case all $G_{\alpha}$'s,
$\alpha < \omega_{1}$, (and not only the very first one,
i.e. $G_{0}$) are indeed Polish. But if the weight of $G$
is greater than $\omega_{1}$, then all $G_{\alpha}$'s, with
$\alpha \geq \omega_{1}$, are non-metrizable.

In order to achieve our final goal and complete the reduction, we
analyze $0$-soft homomorphisms with Polish kernels between
(generally speaking, non-metrizable)
$\text{AE}(0)$-groups. A characterization of such
homomorphisms, which is recorded in Proposition \ref{P:polishkernel},
states that a $0$-soft homomorphism
$f \colon G \to L$ of $\text{AE}(0)$-groups has a Polish kernel
if and only if there exists a pullback diagram
\[
\begin{CD}
G @>f>> L\\
@V{p}VV @VV{q}V\\
G_{0} @>f_{0}>> L_{0},
\end{CD}
\]

\noindent where $G_{0}$ and $L_{0}$ are Polish groups
and the homomorphisms $p \colon G \to G_{0}$ and
$q \colon L \to L_{0}$ are $0$-soft. Theorem
\ref{T:ae0} and Proposition \ref{P:polishkernel}
together complete the required reduction.

Polish groups and their actions have been
extensively studied in a variety of directions (ergodic theory,
group representations,
operator algebras; see \cite{bekech96} for further
discussion and references). Some of the central themes of
the theory of Polish groups counterparts of which (for
arbitrary $\text{AE}(0)$-groups) are considered below  are the
existence of universal groups, the existence of universal actions and
characterization of closed subgroups of the symmetric group $S_{\infty}$.

In Section \ref{S:applications} we use above mentioned spectral
characterizations and present extensions of some of these results
for $\text{AE}(0)$-groups. We prove the existence of  universal
$\text{AE}(0)$-groups of a given weight (Proposition \ref{P:univ})
and the existence of universal actions of
$\text{AE}(0)$-groups of a given weight
on compact $\text{AE}(0)$-spaces of the same
weight (Theorem \ref{T:actionuniv}).

Theorem \ref{T:closed} characterizes $\text{AE}(0)$-groups which
are isomorphic to closed subgroups of powers $S_{\infty}^{\tau}$ of
the symmetric group $S_{\infty}$ - the group
of all bijections of ${\mathbb N}$ under the relative topology
inherited from ${\mathbb N}^{\mathbb N}$. This result extends the
corresponding observation \cite[Theorem 1.5.1]{bekech96} for the
group $S_{\infty}$ itself. As a corollary (Corollary
\ref{C:polish}) we note that if a
Polish group $G$ can be embedded (as a closed subgroup) into
$S_{\infty}^{\tau}$ for some $\tau$, then $G$ can be embedded
into $S_{\infty}$ as well. In light of
\cite{dough94} this shows that there exist zero-dimensional
Polish groups
which can not be embedded into $S_{\infty}^{\tau}$ as closed
subgroups for any cardinal number $\tau$.

Finally we use Theorem \ref{T:ae0} to prove (Theorem \ref{T:baire})
that every $\text{AE}(0)$-group is Baire isomorphic to the product of Polish groups.
In light of Theorem G this result appears to be new even for compact groups.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Preliminaries}\label{S:pre}
All topological spaces below are assumed to be Tychonov (i.e.
completely regular and Hausdorff) and all maps (except
in Subsection \ref{SS:baire}) are continuous. We consider
only Lebesgue dimension $\dim$. Definitions of concepts related to inverse spectra
can be found in \cite{chibook96}. ${\mathbb R}$
denotes the real line and ${\mathbb Q}$ stand for the Hilbert cube.

\subsection{Definitions of $\text{AE}(n)$-spaces and $n$-soft maps}\label{SS:aen}
A comprehensive introduction into general theory
of $\text{AE}(n)$-spaces and $n$-soft maps can be found in \cite{chibook96}.
$C(X)$ denotes the set of all continous real-valued functions defined on $X$.

\begin{defin}\label{D:6.1.3}
A space $X$ is called an absolute  extensor in dimension $n$
(shortly, $\text{AE}(n)$-space), $n=0,1,\dots$, if for each at
most $n$-dimensional space $Z$ and each subspace $Z_0$ of $Z$,
any map $f \colon Z_0 \to X$, such that
$C(f)(C(X)) \subseteq \{ \varphi |Z_{0} \colon \varphi \in C(Z) \}$,
can be extended to $Z$.
\end{defin}

For compact spaces this definition is equivalent to the standard one.

\begin{pro}\label{P:6.1.11}
A compact space $X$ is a $\text{AE}(n)$-space
if and only if for each at most $n$-di\-men\-si\-o\-nal
compactum $Z$ and for each closed subspace $Z_0$ of $Z$,
any map $f \colon Z_0 \to X$ has an extension to  $Z$.
\end{pro}

It is known \cite[Chapter 6]{chibook96} that the
class of metrizable $\text{AE}(0)$-spaces
coincides with the class of Polish spaces. Every
$\text{AE}(0)$-spaces has a countable Suslin number.

\begin{defin}\label{D:6.1.16}
A map $f \colon X \to Y$ between $\text{AE}(n)$-spaces
is $n$-soft if and only if for each at most $n$-dimensional
realcompact space $Z$, for its closed subspace $Z_0$, and for
any two maps $g \colon Z_0 \to X$ and $h \colon Z \to Y$ such that
$f\circ g = h|Z_0$ and
$C(g)(C(X)) \subseteq \{ \varphi |Z_{0} \colon \varphi \in C(Z)\}$,
there exists a map $k \colon Z \to X$ such that
$k|Z_0 = g$ and $f\circ k = h$.
\end{defin}

It is easy to check that $X$ is $\text{AE}(n)$-space if and only if
the constant map $X \to \{ \operatorname{pt}\}$ is $n$-soft.
It is important to note that every $0$-soft map between $\text{AE}(0)$-spaces
is surjective and open (\cite[Lemma 6.1.13 \& Proposition 6.1.26]{chibook96}) and that
for surjections between Polish spaces the converse of this fact is also true.

We say (see \cite[Section 6.3]{chibook96}) that
a map $f \colon X \to Y$ has a Polish kernel if there exists a Polish
space $P$ such that
$X$ is $C$-embedded in the product $Y \times P$ so that $f$ coincides
with the restriction $\pi_{Y}|X$ of the projection
$\pi_{Y} \colon Y \times P \to Y$. Obviously any map between
Polish spaces has a Polish kernel.

\subsection{Set-theoretical preliminaries}\label{SS:set}
Let $A$  be a partially ordered directed set (i.e.
for every two elements  $\alpha ,\beta \in A$  there exists
an element  $\gamma \in A$  such that  $\gamma \geq \alpha$
and  $\gamma \geq \beta$). We say that a subset
$A_1 \subseteq A$ of $A$ majorates another subset
$A_2 \subseteq A$ of $A$ if for each element $\alpha_2 \in A_2$
there exists an element $\alpha_1 \in A_1$ such that
$\alpha_1 \geq \alpha_2$. A subset which majorates $A$
is called cofinal in $A$. A subset of  $A$  is said to
be a chain if every two elements of it are comparable.
The symbol $\sup B$ , where  $B \subseteq A$, denotes the
lower upper bound of $B$ (if such an element exists in $A$).
Let now $\tau$ be an infinite cardinal number. A subset $B$
of $A$  is said to be $\tau$-closed in $A$ if for each chain
$C \subseteq B$, with ${\mid}C{\mid} \leq \tau$, we have
$\sup C \in B$, whenever the element $\sup C$ exists in $A$.
Finally, a directed set $A$ is said to be $\tau$-complete
if for each chain $B$ of elements of $A$ with
${\mid}C{\mid} \leq \tau$, there exists an element
$\sup C$ in $A$.

The standard example of a $\tau$-complete set can be obtained
as follows. For an arbitrary set $A$ let $\exp A$ denote, as usual,
the collection of all subsets of $A$. There is a natural partial
order on $\exp A$: $A_1 \geq A_2$ if and only if $A_1 \supseteq A_2$.
With this partial order $\exp A$ becomes a directed set.
If we consider only those subsets of the set $A$ which have
cardinality $\leq \tau$, then the corresponding subcollection
of $\exp A$, denoted by $\exp_{\tau}A$, serves as a basic
example of a $\tau$-complete set. Proofs of the following
statements can be found in \cite{chibook96}.

\begin{pro}\label{P:3.1.1}
Let  $\{ A_{t} : t \in T \}$ be a collection of $\tau$-closed and
cofinal subsets of a $\tau$-complete set $A$. If
$\mid T\mid \leq \tau$, then the intersection
$\cap \{ A_{t}: t \in T \}$ is also cofinal
(in particular, non-empty) and $\tau$-closed in $A$ .
\end{pro}

\begin{cor}\label{C:3.1.2}
For each subset $B$, with  $\mid B \mid \leq \tau$, of a
$\tau$-complete set $A$ there exists an element $\gamma \in A$
such that  $\gamma \geq \beta$  for each  $\beta \in B$ .
\end{cor}

\begin{pro}\label{P:search}
Let  $A$  be a $\tau$-complete set,
$L \subseteq A^2$, and suppose the following three
conditions are satisfied:
\begin{description}
\item[Existence] For each $\alpha \in A$ there exists
$\beta \in A$  such that  $(\alpha ,\beta ) \in L$.
\item[Majorantness] If  $(\alpha ,\beta ) \in L$  and
$\gamma \geq \beta$, then  $(\alpha ,\gamma ) \in L$.
\item[$\tau$-closeness] Let $\{ \alpha_{t} : t \in T \}$
be a chain in $A$ with $|T| \leq \tau$. If
$(\alpha_{t}, \beta ) \in L$ for some
$\beta \in A$ and each $t \in T$, then
$(\alpha ,\beta ) \in L$ where $\alpha =
\sup \{\alpha_{t} \colon t \in T \}$.
\end{description}
   Then the set of all  $L$-{\em reflexive} elements of
$A$ (an element $\alpha \in A$ is $L$-reflexive if
$(\alpha ,\alpha ) \in L$)  is cofinal and $\tau$-closed in $A$.
\end{pro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Baire sets and Baire isomorphisms}\label{SS:bairesets}
Recall that elements of the $\sigma$-algebra generated
by functionally open subsets of a space $X$ are called Baire sets of $X$.
A map $f \colon X \to Y$ is a Baire map if inverse images of Baire
sets are Baire sets. A bijection $f \colon X \to Y$ is Baire
isomorphism if both $f$ and $f^{-1}$ are Baire maps.

The following statement (\cite[Proposition 2.5]{chi994}) is
used in the proof of Theorem \ref{T:baire}.

\begin{pro}\label{P:bb}
Let $f \colon X \to Y$ be a $0$-soft map of
$\text{AE}(0)$-spaces. Then there exists a Baire map
$g \colon Y \to X$ such that $f \circ g = \operatorname{id}_{Y}$.
\end{pro}

If $f$ is a continuous homomorphism of Polish groups, then
the existence of such a $g$ has been observed by Dixmier
\cite[Theorem 1.2.4]{bekech96}, \cite[Theorem 12.17]{kech95}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{$\text{AE}(0)$-groups and actions of
$\text{AE}(0)$-groups -- spectral representations}\label{S:spectral}

In this section we present spectral characterizations of
$\text{AE}(0)$-groups. We also present a spectral description
of actions of $\text{AE}(0)$-groups on $\text{AE}(0)$-spaces.

\begin{pro}\label{P:B8.2.1}
Each  topological $\text{AE}(0)$-group $G$ is topologically
and algebraically isomorphic to the limit of a factorizing
$\omega$-spectrum ${\mathcal S}_{G} = \{ G_{\alpha},
p_{\alpha}^{\beta}, A\}$
consisting of Polish groups $G_{\alpha}$, $\alpha \in A$, and $0$-soft
limit homomorphisms $p_{\alpha} \colon G \to G_{\alpha}$, $\alpha \in A$.
In particular, all short projections
$p_{\alpha}^{\beta} \colon G_{\beta} \to G_{\alpha}$,
$\alpha \leq \beta$, $\alpha ,\beta \in A$, are $0$-soft homomorphisms.
\end{pro}
\begin{proof}
By \cite[Theorem 6.3.2 or Proposition 6.3.5]{chibook96},
the space $G$ can be
represented as the limit space of a factorizing $\omega$-spectrum
${\mathcal S} = \{ G_{\alpha}, p_{\alpha}^{\beta}, \widetilde{A}\}$
consisting
of Polish spaces (i.e. $\text{AE}(0)$-spaces of countable weight)
and $0$-soft limit
projections. Let us show that this spectrum contains
$\omega$-closed and cofinal subspectrum consisting of
topological groups and limit projections that are
(continuous) homomorphisms.

Let $\mu \colon G \times G \to G$ and $\nu \colon G \to G$ be
continuous operations of multiplication and inversion given on
$G$ as on a topological group. We apply \cite[Theorem 1.3.6]{chibook96} to
both $\mu$ and $\nu$. First consider the multiplication.
Clearly, $G \times G$ is the limit space
of the spectrum\\
\[ {\mathcal S} \times {\mathcal S} =
\{ G_{\alpha} \times G_{\alpha}, p_{\alpha}^{\beta}
\times p_{\alpha}^{\beta}, \widetilde{A} \} .\]

\noindent All projections of the spectrum
${\mathcal S} \times {\mathcal S}$ are $0$-soft, and hence,
by \cite[Proposition 6.1.26]{chibook96}, open. The Suslin number
of the product $G \times G$ is obviously countable
(see \cite[proposition 6.1.8]{chibook96}). Consequently,
by \cite[Proposition 1.3.3]{chibook96}, the spectrum
${\mathcal S} \times {\mathcal S}$ is factorizing. Next
we apply \cite[Theorem 1.3.6]{chibook96} to the spectra
${\mathcal S} \times {\mathcal S}$, $\mathcal S$ and to the
map $\mu$ between their limit spaces. Then we get a
$\omega$-closed and cofinal subset $A_{\mu}$ of $\widetilde{A}$
such that for
each $\alpha \in A_{\mu}$ there exists a continuous map
$\mu _{\alpha} \colon G_{\alpha} \times G_{\alpha} \to G_{\alpha}$
such that the diagram\\

\[
\begin{CD}
G \times G @>\mu>> G\\
@V{p_{\alpha}\times p_{\alpha}}VV @VV{p_{\alpha}}V\\
G_{\alpha}\times G_{\alpha} @>\mu_{\alpha}>> G_{\alpha}
\end{CD}
\]
\bigskip

\noindent commutes. In other words,
$p_{\alpha}\circ \mu = \mu _{\alpha}\circ (p_{\alpha} \times p_{\alpha})$
for each $\alpha \in A_{\mu}$.

Next consider a continuous inversion $\nu \colon G \to G$. By
\cite[Theorem 1.3.6]{chibook96}, applied to $\nu$ and the
spectrum ${\mathcal S}$,
there exists a $\omega$-closed and cofinal subset
$A_{\nu}$ of $\widetilde{A}$ such that for
each $\alpha \in A_{\nu}$ there exists a continuous map
$\nu _{\alpha} \colon G_{\alpha}  \to G_{\alpha}$
such that the diagram\\

\[
\begin{CD}
G  @>\nu>> G\\
@V{p_{\alpha}}VV @VV{p_{\alpha}}V\\
G_{\alpha} @>\nu_{\alpha}>> G_{\alpha}
\end{CD}
\]
\bigskip

\noindent commutes. In other words,
$p_{\alpha}\circ \nu = \nu_{\alpha} \circ p_{\alpha}$
for each $\alpha \in A_{\nu}$.

By Proposition \ref{P:3.1.1}, the intersection
$A = A_{\mu} \cap A_{\nu}$ is still $\omega$-closed and cofinal
in $\widetilde{A}$. This guarantees that $G$ is topologically
and algebraically isomorphic to the limit of the factorizing
$\omega$-spectrum
${\mathcal S}_{G} = \{ G_{\alpha}, p_{\alpha}^{\beta} , A\}$.
Also for each $\alpha \in A$ we have two maps
\[ \mu_{\alpha} \colon G_{\alpha} \times G_{\alpha} \to
G_{\alpha}\;\;\text{and}\;\; \nu_{\alpha} \colon G_{\alpha}
\to G_{\alpha} \]

\noindent which allow us to define

\begin{itemize}
\item[(a)]
a continuous multiplication operation on
$G_{\alpha}$ by letting\\
\[ x_{\alpha}\cdot y_{\alpha} = \mu _{\alpha}(x_{\alpha},
y_{\alpha})\;\;
\text{for each}\;\; (x_{\alpha},y_{\alpha}) \in G_{\alpha}
\times G_{\alpha};\]
\medskip
\item[(b)]
a continuous inversion on $G_{\alpha}$ by letting

\[ x_{\alpha}^{-1} = \nu_{\alpha}(x_{\alpha}) \;\;
\text{for each}\;\; x_{\alpha} \in G_{\alpha} .\]
\medskip
\end{itemize}

It is easy to see that $G_{\alpha},\; \alpha \in A$, becomes a
topological group
with respect to these operations. Moreover, for each $\alpha \in A$
the limit projection $p_{\alpha} \colon G \to G_{\alpha}$ becomes a
homomorphism with respect to the above defined operations.
\end{proof}

\begin{cor}\label{C:dim}
Let $G$ be a $\text{AE}(0)$-group such that $\dim G \leq n$. Then
$G$ is topologically
and algebraically isomorphic to the limit of a factorizing
$\omega$-spectrum ${\mathcal S}_{G} = \{ G_{\alpha},
p_{\alpha}^{\beta}, A\}$
consisting of at most $n$-dimensional Polish groups $G_{\alpha}$,
$\alpha \in A$, and $0$-soft
limit homomorphisms $p_{\alpha} \colon G \to G_{\alpha}$, $\alpha \in A$.
\end{cor}
\begin{proof}
By Proposition \ref{P:B8.2.1}, $G = \lim {\mathcal S}_{1}$,
where ${\mathcal S}_{1} = \{ G_{\alpha}, p_{\alpha}^{\beta}, A_{1}\}$
is a factorizing $\omega$-spectrum consisting of Polish groups
and $0$-soft limit homomorphisms. By \cite[Theorem 1.3.10]{chibook96},
$G = \lim {\mathcal S}_{2}$, where
${\mathcal S}_{2} = \{ G_{\alpha}, p_{\alpha}^{\beta}, A_{2}\}$
is a factorizing $\omega$-spectrum consisting of at most
$n$-dimensional Polish spaces. It follows directly from the
proofs of Proposition \ref{P:B8.2.1} and
\cite[Theorem 1.3.10]{chibook96}, that both indexing sets
$A_{1}$ and $A_{2}$ can be assumed to be closed and
$\omega$-complete subsets of a $\omega$-complete set $B$.
By Proposition
\ref{P:3.1.1}, $A = A_{1} \cap A_{2}$ is still cofinal and
$\omega$-closed subset of $B$.
Consequesntly, the factorizing $\omega$-spectrum
${\mathcal S}_{G} = \{ G_{\alpha}, p_{\alpha}^{\beta}, A\}$
consists of at most $n$-dimensional Polish groups and $0$-soft
limit homomorphisms. It only remains to note that
$G = \lim{\mathcal S}_{G}$.
\end{proof}

\begin{cor}\label{C:emb}
Let $\tau \geq \omega$. Every $\text{AE}(0)$-group of
weight $\tau \geq \omega$
is topologically and algebraically isomorphic to a closed
and $\text{C}$-embedded
subgroup of the product
$\displaystyle \prod\{ G_{t} \colon t \in T\}$, where $G_{t}$,
$t \in T$, is a Polish group and $|T| = \tau$.
\end{cor}
\begin{proof}
Let $G$ be an $\text{AE}(0)$-group of weight $\tau$.
If $\tau = \omega$, then
$G$ is Polish (see \cite[Chapter 6]{chibook96} and consequently
there is nothing to prove.

If $\tau > \omega$, then by Proposition \ref{P:B8.2.1}, $G$ is
topologically and algebraically
isomorphic to the limit of a factorizing $\omega$-spectrum
${\mathcal S}_{G} = \{ G_{t}, p_{t}^{t^{\prime}}, T\}$
with $|T|  = \tau$.
Clearly $\lim {\mathcal S}_{G}$ is isomorphic to a closed subgroup
of the product
$\displaystyle \prod\{ G_{t}, t \in A\}$. Since the spectrum
${\mathcal S}_{G}$ is factorizing,
it follows that $\lim {\mathcal S}_{G}$ is $\text{C}$-embedded
in $\displaystyle \prod\{ G_{t}, t \in T\}$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%

\begin{thm}\label{T:ae0}
Let $G$ be a topological group of weight $\tau > \omega$.
Then the following conditions are equivalent:
\begin{itemize}
\item[(a)]
$G$ is a $\text{AE}(0)$-group.
\item[(b)]
There exists a well-ordered inverse spectrum
${\mathcal S}_{G} = \{ G_{\alpha},
p_{\alpha}^{\alpha +1},\tau \}$ satisfying the following properties:
\begin{enumerate}
\item
$G_{\alpha}$ is a $\text{AE}(0)$-group and $p^{\alpha +1}_{\alpha}
\colon G_{\alpha +1} \to G_{\alpha}$ is a $0$-soft
homomorphism with Polish kernel, $\alpha < \tau$.
\item
If $\beta < \tau$ is a limit ordinal, then the diagonal product
\[\triangle\{ p_{\alpha}^{\beta} \colon \alpha < \beta\}
\colon G_{\beta} \to \lim\{ G_{\alpha},
p_{\alpha}^{\alpha+1}, \alpha < \beta \}\]
is a topological and algebraic isomorphism.
\item
$G$ is topologically and algebraically isomorphic to
$\lim {\mathcal S}_{G}$.
\item
$G_{0}$ is a Polish group.
\end{enumerate}
\end{itemize}
\end{thm}
\begin{proof}
(a) $\Longrightarrow$ (b). By Corollary \ref{C:emb}, we may
assume that $G$ is a closed and $\text{C}$-embedded subgroup
of the product $\displaystyle \prod\{ X_{a} \colon a \in A\}$,
$|A| = \tau$, of Polish groups $X_{a}$, $a \in A$. There exists
a proper, functionally closed and $0$-invertible map
$f \colon Y \to \prod\{ X_{a} \colon a \in A \}$, where $Y$
is a spectrally complete (see \cite[p.247]{chibook96})
realcompact space of weight $\tau$ and dimension $\dim Y = 0$
(see \cite[Proposition 6.2.13]{chibook96} for details).
Consider the inverse image $f^{-1}(G) \subseteq Y$ of $G$ and
the map $f|f^{-1} \colon f^{-1}(G) \to G$. Since $G$ is
$\text{C}$-embedded in the product $\displaystyle \prod\{
X_{a} \colon a \in A\}$, since $\dim Y = 0$ and since $G$,
according to (a), is an $AE(0)$-space, there exists a map
$g \colon Y \to G$ such that $g|f^{-1}(G) = f|f^{-1}(G)$.

Next let us denote by
\[ \pi_{B} \colon \prod\{ X_{a} \colon a \in A\} \to
\prod\{ X_{a} \colon a \in B\}\]
and
\[\pi^{B}_{C} \colon \prod\{ X_{a} \colon a \in B\} \to
\prod\{ X_{a} \colon a \in C \}\]
the natural projections onto the corresponding subproducts
($C \subseteq B \subseteq A$).
We call a subset $B \subseteq A$ admissible (compare
with the proof of \cite[Theorem 6.3.1]{chibook96}) if the
following equality
\[ \pi_{B}(g(f^{-1}(x))) = \pi_{B}(x)\]
is true for each point $x \in \pi_{B}^{-1}\left(\pi_{B}(G)
\right)$. We need the following properties of admissible sets.

{\em Claim 1. The union of arbitrary collection of admissible
sets is admissible}.

Indeed let $\{ B_{t} \colon t \in T\}$ be a collection of
admissible sets and $B = \cup \{ B_{t} \colon t \in T\}$.
Let $x \in \pi_{B}^{-1}\left(\pi_{B}(G)\right)$. Clearly
$x \in \pi_{B_{t}}^{-1}\left(\pi_{B_{t}}(G)\right)$ for each
$t \in T$ and consequently
\[ \pi_{B_{t}}(g(f^{-1}(x))) = \pi_{B_{t}}(x) \;\;\text{for
each}\;\; t \in T .\]
Obviously, $\pi_{B}(x) \in \pi_{B}(g(f^{-1}(x)))$ and it
therefore suffices to show that the set $\pi_{B}(g(f^{-1}(x)))$
contains only one point. Assuming that there is a point
$y \in \pi_{B}(g(f^{-1}(x)))$ such that $y \neq \pi_{B}(x)$
we conclude (having in mind that $B = \cup\{ B_{t} \colon
t \in T\}$) that there must be an index $t \in T$ such that
$\pi_{B_{t}}^{B}(y) \neq \pi_{B_{t}}^{B}\left(\pi_{B}(x)\right)$.
But this is impossible
\[ \pi_{B_{t}}^{B}(y) \in \pi_{B_{t}}^{B}\left(\pi_{B}
(g(f^{-1}(x)))\right) = \pi_{B_{t}}(g(f^{-1}(x))) = \pi_{B_{t}}(x) = \pi_{B_{t}}^{B}\left(\pi_{B}(x)\right) .\]

{\em Claim 2. If $B \subseteq A$ is admissible, then the
restriction $\pi_{B}|G \colon G \to \pi_{B}(G)$ is $0$-soft}.

Let $\varphi \colon Z \to \pi_{B}(G)$ and
$\varphi_{0} \colon Z_{0} \to G$ be two maps defined on a
realcompact space $Z$, with $\dim Z = 0$, and its closed
subset $Z_{0}$ respectively. Assume that
$\pi_{B}\varphi_{0} = \varphi |Z_{0}$ and
$C(\varphi_{0})(C(G)) \subseteq C(Z)|Z_{0}$. We wish to
construct a map $\phi \colon Z \to G$ such that
$\phi |Z_{0} = \varphi_{0}$ and $\pi_{B}\phi = \varphi$, i.e.
$\phi$ makes the diagram

\begin{picture}(300,150)
\put(80,110){$Z$}
\put(80,10){$Z_{0}$}
\put(80,100){$\cap$}
\put(87,102){\vector(0,-1){79}}
\put(65,60){$\operatorname{incl}$}
\put(95,17){\vector(1,1){90}}
\put(95,112){\vector(1,0){90}}
\put(95,12){\vector(1,0){80}}
\put(190,110){$G$}
\put(180,10){$\pi_{B}(G)$}
\put(195,107){\vector(0,-1){83}}
\put(135,118){$\varphi_{0}$}
\put(135,17){$\varphi$}
\put(135,67){$\phi$}
\put(198,60){$\pi_{B}|G$}
\end{picture}

\noindent commutative. Since, according to our choice,
all $X_{a}$'s are $\text{AE}(0)$-spaces (recall that each
$X_{a}$ is a Polish space), so is the product
$\prod\{ X_{a} \colon a \in A-B\}$. This implies the $0$-softness
of the projection $\pi_{B}$ and hence of its restriction
\[ \pi_{B}|\pi_{B}^{-1}\left(\pi_{B}(G)\right) \colon
\pi_{B}^{-1}\left(\pi_{B}(G)\right) \to \pi_{B}(G) .\]
Then there exists a map
$\phi^{\prime\prime} \colon Z \to \pi_{B}^{-1}\left(\pi_{B}(G)\right)$
such that $\phi^{\prime\prime}|Z_{0} = \varphi_{0}$ and
$\pi_{B}\phi^{\prime\prime} = \varphi$. Since $f$ is
$0$-invertible (and $\dim Z = 0$), there exists a map
$\phi^{\prime} \colon Z \to Y$ such that
$f\phi^{\prime} = \phi^{\prime\prime}$. Now let
$\phi = g\phi^{\prime}$. Since $g|f^{-1}(G) = f|f^{-1}(G)$,
we have $\varphi_{0} = \phi |Z_{0}$. Finally observe that
the admissibility of $B$ implies $\varphi = \pi_{B}\phi$ as required.

{\em Claim 3. For each countable subset $C \subseteq A$
there exists a countable admissible subset $B \subseteq A$
such that $C \subseteq B$}.

Since $w(Y) = \tau$ and $\dim Y = 0$, it follows
(consult \cite[Theorem 1.3.10]{chibook96}) that $Y$
can be represented as the limit space of a factorizing
$\omega$-spectrum ${\mathcal S}_{Y} = \{ Y_{B}, q^{B}_{C},
\exp_{\omega}A \}$ consisting of zero-dimensional Polish
spaces $Y_{B}$, $B \in \exp_{\omega}A$, and continuous
surjections $q_{C}^{B} \colon Y_{B} \to Y_{C}$, $C \subseteq B$,
$C,B \in \exp_{\omega}A$. Consider also the standard factorizing
$\omega$-spectrum $\displaystyle {\mathcal S}_{X} =
\{ \prod\{ X_{a} \colon a \in B\} ,\pi_{C}^{B}, \exp_{\omega}A \}$
consisting of countable subproducts of the product
$\displaystyle \prod\{ X_{a} \colon a \in A \}$ and corresponding
natural projections. Obviously the full product coincides
with the limit of ${\mathcal S}_{X}$. One more factorizing
$\omega$-spectrum arises naturally. This is the spectrum
${\mathcal S}_{G} = \{ \pi_{B}(G), \pi_{C}^{B}|\pi_{B}(G),
\exp_{\omega}A \}$ the limit of which coincides with $G$.

Consider the map $f \colon \lim {\mathcal S}_{Y} \to
\lim {\mathcal S}_{X}$. By \cite[Theorem 1.3.4]{chibook96},
there is a cofinal and $\omega$-closed subset
${\mathcal T}_{f}$ of $\exp_{\omega}A$ such that for each
$B \in {\mathcal T}_{f}$ there is a map
$f_{B} \colon Y_{B} \to \prod\{ X_{a} \colon a \in B \}$ such
that $f_{B}\circ q_{B} = \pi_{B}\circ f$. Moreover, these maps
form a morphism
\[ \{ f_{B} ; B \in {\mathcal T}_{f}\} \colon {\mathcal S}_{Y}
\to {\mathcal S}_{X}\]
limit of which coincides with $f$. Since $f$ is proper and
functionally closed, we may assume (see
\cite[Proposition 6.2.9]{chibook96}) without loss of generality
(considering a smaller cofinal and $\omega$-subset of
${\mathcal T}_{f}$ if necessary) that the above indicated morphism
is bicommutative. This simply means that
$q_{B}f^{-1}(K) = f_{B}^{-1}\left(\pi_{B}(K)\right)$ for any
$B \in {\mathcal T}_{f}$ and any closed subset $K$ of the product
$\displaystyle \prod\{ X_{a} \colon a \in A\}$.

Similarly, applying \cite[Theorem 1.3.4]{chibook96} to the
map $g \colon \lim {\mathcal S}_{Y} \to \lim {\mathcal S}_{G}$,
we obtain a cofinal and $\omega$-closed subset
${\mathcal T}_{g}$ of $\exp_{\omega}A$ and the associated to
it morphism
\[ \{ g_{B} \colon Y_{B} \to \pi_{B}(G) ; B \in
{\mathcal T}_{g} \} \colon {\mathcal S}_{Y} \to {\mathcal S}_{G}\]

\noindent limit of which coincides with the map $g$.

By Proposition \ref{P:3.1.1}, the intersection
${\mathcal T} = {\mathcal T}_{f} \cap {\mathcal T}_{g}$
is still a cofinal and $\omega$-closed subset of $\exp_{\omega}A$.
It therefore suffices to show that each $B \in {\mathcal T}$ is
an admissible subset of $A$. Consider a point
$x \in \pi_{B}^{-1}\left( \pi_{B}(G)\right)$. First observe
that the bicommutativity of the morphism associated with
${\mathcal T}_{f}$ implies that
$q_{B}(f^{-1}(x)) = f_{B}^{-1}(\pi_{B}(x))$. Since the
maps $f_{B}$ and $g_{B}$ coincide on $f_{B}^{-1}(\pi_{B}(G))$ we have
\begin{multline*}
\pi_{B}(g(f^{-1}(x))) = g_{B}(q_{B}(f^{-1}(x))) =
g_{B}(f_{B}^{-1}(\pi_{B}(x))) =\\ f_{B}(f_{B}^{-1}(\pi_{B}(x))) =
\pi_{B}(x)
\end{multline*}
as required.

{\em Claim 4. If $C$ and $B$ are admissible subsets of $A$
and $C \subseteq B$, then the map
$\pi_{C}^{B}|\pi_{B}(G) \colon \pi_{B}(G) \to \pi_{C}(G)$
is $0$-soft}.

This property follows from Claim 2 and \cite[Lemma 6.1.15]{chibook96}.

After having all the needed properties of admissible subsets
established we proceed as follows. Since $|A| = \tau$ we can
write $A = \{ a_{\alpha} \colon \alpha < \tau\}$. By Claim 3,
each $a_{\alpha} \in A$ is contained in a countable admissible
subset $B_{\alpha} \subseteq A$. Let
$A_{\alpha} = \cup\{ B_{\beta} \colon \beta \leq \alpha\}$.
We use these sets to define a transfinite inverse spectrum
${\mathcal S} = \{ G_{\alpha}, p_{\alpha}^{\alpha +1}, \tau \}$
as follows. Let $G_{\alpha} = \pi_{A_{\alpha}}(G)$ and
$p_{\alpha}^{\alpha +1} = \pi_{A_{\alpha}}^{A_{\alpha +1}}|
G_{\alpha +1}$ for each $\alpha < \tau$. All the required
properties of the spectrum ${\mathcal S}_{G}$ are satisfied by
construction.

The implication (b) $\Longrightarrow$ (a) immediately
follows from \cite[Proposition 6.3.4]{chibook96}
\end{proof}

\begin{rem}
Actually $0$-soft homomorphism $p_{\alpha}^{\alpha +1} \colon
G_{\alpha+1} \to G_{\alpha}$, $\alpha < \tau$, in Theorem \ref{T:ae0}(b)1
has Polish kernel in a somewhat
 stonger sense than the original definition  presented in
Subsection \ref{SS:aen}. Namely a Polish space $P$ (from
the definition), such that $G_{\alpha +1}$ admist a
$C$-embedding into the product $G_{\alpha} \times P$ in such a way that
$p_{\alpha}^{\alpha +1}$ coincides with the restriction of
the projection
$\pi_{G_{\alpha}} \colon G_{\alpha} \times P \to G_{\alpha}$,
can be chosen to be a Polish group and the embedding of
$G_{\alpha +1} \to G_{\alpha} \times P$ can be assumed to
be a homomorphism. Of course this implies that
$\ker p_{\alpha}^{\alpha +1}$, as a closed subgroup of $P$, is
itself a Polish group. It would be interesting to see whether
the converse of this observation is also true, i.e. is it true that
if the kernel $\ker p$ of a $0$-soft homomorphism $p \colon G \to L$
of $\text{AE}(0)$-groups is Polish, then $p$ has a Polish kernel
in the sense of Subsection \ref{SS:aen}.
\end{rem}

Next we characterize $0$-soft homomorphisms of $\text{AE}(0)$-groups
with Polish kernels.

\begin{pro}\label{P:polishkernel}
A $0$-soft homomorphism $f \colon G \to L$ between $\text{AE}(0)$-groups
has a Polish kernel
if and only if there exist factorizing $\omega$-spectra\;
${\mathcal S}_{G} = \{ G_{\alpha}, p_{\alpha}^{\beta}, A \}$,
${\mathcal S}_{L} = \{ L_{\alpha}, q_{\alpha}^{\beta}, A \}$,
consisting of Polish groups and $0$-soft limit homomorphisms, and
a morphism
$\{ f_{\alpha} \} \colon {\mathcal S}_{G} \to {\mathcal S}_{L}$,
consisting of $0$-soft homomorphisms, such that the following
conditions are satisfied:
\begin{enumerate}
\item
$G = \lim{\mathcal S}_{G}$, $L = \lim{\mathcal S}_{L}$ and
$f = \lim \{ f_{\alpha} \}$.
\item
All limit projections of the spectra ${\mathcal S}_{G}$ and
${\mathcal S}_{L}$ are $0$-soft.
\item
All limit square diagrams, generated by limit projections of
spectra ${\mathcal S}_{G}$ and ${\mathcal S}_{L}$, by elements
of the morphism $\{ f_{\alpha} \}$ and by the map $f$, are the
Cartesian squares.
\end{enumerate}
\end{pro}
\begin{proof}
By Proposition \ref{P:B8.2.1}, we may assume, without loss of
generality, that
both $G$ and $L$ are topologically and algebraically isomorphic
to the limits of factorizing $\omega$-spectra
${\mathcal S}_{G} = \{ G_{\alpha}, p_{\alpha}^{\beta}, A \}$
and ${\mathcal S}_{L} = \{ L_{\alpha}, q_{\alpha}^{\beta}, A \}$,
consisting of Polish groups and $0$-soft limit homomorphisms.
By \cite[Theorem 1.3.6]{chibook96}, we may also assume that the
map
$f$ is the limit of a morphism
$\{ f_{\alpha} \colon G_{\alpha} \to L_{\alpha} ; A\}$,
consisting of continuous maps
$f_{\alpha} \colon G_{\alpha} \to L_{\alpha}$.
Since $f$ itself and all the limit projections
$p_{\alpha} \colon G \to G_{\alpha}$ and
$q_{\alpha} \colon L \to L_{\alpha}$ are homomorphisms
between respective groups, it follows easily that
$f_{\alpha} \colon G_{\alpha} \to L_{\alpha}$ is
also a homomorphism, $\alpha \in A$. Finally since $f$ is
$0$-soft and has a Polish kernel, it follows, by
\cite[Theorem 6.3.1(vi)]{chibook96}, that all limit square diagrams

\[
\begin{CD}
G @>f>> L\\
@V{p_{\alpha}}VV @VV{q_{\alpha}}V\\
G_{\alpha} @>f_{\alpha}>> L_{\alpha},
\end{CD}
\]
\bigskip

\noindent generated by limit projections of
spectra ${\mathcal S}_{G}$ and ${\mathcal S}_{L}$, by elements
of the morphism $\{ f_{\alpha} \}$ and by the map $f$, are the
Cartesian squares.
\end{proof}

Below, in Subsection \ref{SS:universal}, we consider actions of
$\text{AE}(0)$-groups on $\text{AE}(0)$-spaces. Main tool here
is the following statement (see \cite[Theorem 8.7.1]{chibook96}).

\begin{pro}\label{P:action}
Let $\lambda \colon G \times X \to X$ be a continuous action of
an $\text{AE}(0)$-group $G$
on an $\text{AE}(0)$-space $X$. Suppose that $X$ is homeomorphic
to the limit space of a factorizing $\omega$-spectrum
${\mathcal S}_{X} = \{ X_{\alpha}, p_{\alpha}^{\beta}, A\}$
consisting of Polish spaces and $0$-soft limit projections.
Suppose also that $G$ is topologically and algebraically
isomorphic to the limit of the factorizing $\omega$-spectrum
${\mathcal S}_{G} = \{ G_{\alpha}, s_{\alpha}^{\beta}, A\}$
consisting of Polish groups and $0$-soft limit homomorphisms.
Then $\lambda$ is the limit of ``level actions", i.e.
$\lambda = \lim \lambda_{\alpha}$, where
\[ \{ \lambda_{\alpha} \colon G_{\alpha} \times X_{\alpha}
\to X_{\alpha} , B\} \colon {\mathcal S}_{G}|B \times
{\mathcal S}_{X}|B \to {\mathcal S}_{X}|B\]
 \noindent is a morphism between the spectra
${\mathcal S}_{G}|B \times {\mathcal S}_{X}|B$ and
${\mathcal S}_{X}|B$ and $B$ is a cofinal and
$\omega$-closed subset of the indexing set $A$.
\end{pro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

\section{Applications}\label{S:applications}

\subsection{Universal $\text{AE}(0)$-groups and universal
actions of $\text{AE}(0)$-groups}\label{SS:universal}
In this Subsection we prove the existence of universal
$\text{AE}(0)$-groups of a given weight as well as the
existence of a universal action of a $\text{AE}(0)$-group
of a given weight on a compact $\text{AE}(0)$-space of the
same weight.

\begin{pro}\label{P:univ}
Let $\tau \geq \omega$. The class of $\text{AE}(0)$-groups of weight
$\leq \tau$ contains a universal element. More formally, every
$\text{AE}(0)$-group is topologically and algebraically isomorphic
to a closed and $\text{C}$-embedded subgroup of the power
$\left(\operatorname{Aut}({\mathbb Q})\right)^{\tau}$,
where $\operatorname{Aut}({\mathbb Q})$
denotes the group of autohomeomorphisms of the Hilbert cube ${\mathbb Q}$.
\end{pro}
\begin{proof}
Let $G$ be a $\text{AE}(0)$-group of weight $\tau$. By
Corollary \ref{C:emb}, $G$ is topologically and algebraically
isomorphic to a closed and $C$-embedded subgroup of the
product $\displaystyle \prod\{ G_{t} \colon t \in T\}$,
where $G_{t}$ is a Polish group for each $t \in T$ and $|T| = \tau$.
By Uspenskii's theorem \cite{usp86}, $G_{t}$ can be identified with a closed
subgroup of $\operatorname{Aut}({\mathbb Q})$. Obviously
$\displaystyle \prod\{ G_{t} \colon t \in T\}$,
and consequently $G$, is a closed and
$C$-embedded subgroup of
$\left(\operatorname{Aut}({\mathbb Q})\right)^{\tau}$.
\end{proof}

Let $\tau > \omega$. Clearly the $\text{AE}(0)$-group
$\left(\operatorname{Aut}({\mathbb Q})\right)^{\tau}$ (i.e. the $\tau$-th power
of the group $\operatorname{Aut}({\mathbb Q})$)
continuously acts on the Tychonov cube ${\mathbb Q}^{\tau}$
via the natural action (``coordinatewise evaluation")
\[ \operatorname{ev}_{\tau} \colon \left(\operatorname{Aut}
({\mathbb Q})\right)^{\tau} \times
{\mathbb Q}^{\tau} \to {\mathbb Q}^{\tau} ,\]

\noindent which is defined by letting
\begin{multline*}
\operatorname{ev}_{\tau}\left( \{ g_{\alpha} \colon \alpha < \tau \} ,
\{ q_{\alpha} \colon \alpha < \tau\}\right) =
\left\{ g_{\alpha}(q_{\alpha}) \colon \alpha < \tau \right\}
\;\;\text{for each}\\
 \left( \{ g_{\alpha} \colon \alpha < \tau \} ,
\{ q_{\alpha} \colon \alpha < \tau\}\right) \in
\left(\operatorname{Aut}({\mathbb Q})\right)^{\tau} \times
{\mathbb Q}^{\tau} .
\end{multline*}

\begin{thm}\label{T:actionuniv}
Let $\tau > \omega$. The action $\operatorname{ev}_{\tau} \colon
\left(\operatorname{Aut}({\mathbb Q})\right)^{\tau} \times
{\mathbb Q}^{\tau} \to {\mathbb Q}^{\tau}$ is universal
in the category of actions of $\text{AE}(0)$-groups
of weight $\tau$ on  compact $\text{AE}(0)$-spaces of weight
$\tau$. More formally,
let $\lambda \colon G \times X \to X$ be a continuous action of a
$\text{AE}(0)$-group $G$ of weight $\tau$ on a compact
$\text{AE}(0)$-space $X$ of weight
$\tau$. Then there exists a topological and algebraic embedding
$i_{G} \colon G \to \left(\operatorname{Aut}
({\mathbb Q})\right)^{\tau}$ with a closed image and an embedding
$i_{X} \colon X \to {\mathbb Q}^{\tau}$ such that the following diagram
\[
\begin{CD}
\left(\operatorname{Aut}
({\mathbb Q})\right)^{\tau}\times {\mathbb Q}^{\tau}
@>\operatorname{ev}_{\tau}>> {\mathbb Q}^{\tau}\\
@A{i_{G}}\times i_{X}AA @AA{i_{X}}A\\
G \times X @>\lambda>> X
\end{CD}
\]

\noindent commutes.
\end{thm}
\begin{proof}
By Proposition \ref{P:B8.2.1},  $G$ can be represented as
the limit of a factorizing $\omega$-spectrum
${\mathcal S}_{G} = \{ G_{\alpha}, s_{\alpha}^{\beta}, A\}$
consisting of Polish groups and $0$-soft limit homomorphisms.
Similarly, by \cite[Proposition 6.3.5]{chibook96}, $X$ can be
represented as the limit space of a factorizing $\omega$-spectrum
${\mathcal S}_{X} = \{ X_{\alpha}, p_{\alpha}^{\beta}, A\}$
consisting of metrizable compacta and $0$-soft limit projections.
Without loss of generality we may assume that these spectra
${\mathcal S}_{G}$ and ${\mathcal S}_{X}$ have the same indexing
set $A$ and $|A| = \tau$. By Proposition \ref{P:action}, the given
action $\lambda \colon G \times X \to X$ is the limit of level
actions, i.e.
$\lambda = \lim \lambda_{\alpha}$, where
\[ \{ \lambda_{\alpha} \colon G_{\alpha} \times X_{\alpha}
\to X_{\alpha} , B\} \colon {\mathcal S}_{G}|B \times
{\mathcal S}_{X}|B \to {\mathcal S}_{X}|B\]
 \noindent is a morphism between the spectra
${\mathcal S}_{G}|B \times {\mathcal S}_{X}|B$ and
${\mathcal S}_{X}|B$ and $B$ is a cofinal and
$\omega$-closed subset of the indexing set $A$. We may also
assume that $|B| = \tau$.

Since $G = \lim {\mathcal S}_{G}|B$ it follows that the
diagonal product

\[ s = \triangle \{ s_{\alpha} \colon G \to G_{\alpha} ,
\alpha \in B\} \colon G \to \prod\{ G_{\alpha} \colon \alpha \in B\}\]

\noindent is a topological and algebraic isomorphism with
a closed image. Similarly
the diagonal product

\[ p = \triangle \{ p_{\alpha} \colon X \to X_{\alpha} ,
\alpha \in B\} \colon X \to \prod\{ X_{\alpha} \colon \alpha \in B\}\]

\noindent is an embedding. Consider also the product action
\[ \widetilde{\lambda} \colon \prod\{ G_{\alpha} \colon \alpha \in B\}
\times \prod\{ X_{\alpha} \colon \alpha \in B\} \to
\prod\{ X_{\alpha} \colon \alpha \in B\}\]

\noindent defined by letting

\begin{multline*}
 \widetilde{\lambda}\left( \{ g_{\alpha} \colon
\alpha \in B\} ,\{ x_{\alpha} \colon \alpha \in B \}\right) =
\{ \lambda_{\alpha}(g_{\alpha}, x_{\alpha}) \colon \alpha \in B\}
\;\;\text{for each}\\
 \left( \{ g_{\alpha} \colon
\alpha \in B\} ,\{ x_{\alpha} \colon \alpha \in B \}\right)
\in \prod\{ G_{\alpha} \colon \alpha \in B\} \times
\prod\{ X_{\alpha} \colon \alpha \in B\} .
\end{multline*}

\noindent Note that
$\widetilde{\lambda}\circ (s\times p) = p \circ \lambda$,
i.e. the following diagram

\[
\begin{CD}
\prod\{ G_{\alpha} \colon \alpha \in B\} \times
\prod\{ X_{\alpha} \colon \alpha \in B\} @>\widetilde{\lambda}>>
\prod\{ X_{\alpha} \colon \alpha \in B\}\\
@A{s\times p}AA @AA{p}A\\
G\times X @>\lambda>> X
\end{CD}
\]

\noindent is commutative.

Since for each $\alpha \in B$ the group $G_{\alpha}$ is Polish, it follows
by \cite{megr96} (see also \cite[Theorem 2.6.7]{bekech96}) that
there exist a
topological and algebraic embedding $j_{\alpha} \colon G_{\alpha} \to \operatorname{Aut}({\mathbb Q}_{\alpha})$
with a closed image and an embedding
$i_{\alpha} \colon X_{\alpha} \to {\mathbb Q}_{\alpha}$
(here ${\mathbb Q}_{\alpha}$ denotes a copy of the
Hilbert cube ${\mathbb Q}$) such that
$\operatorname{ev}_{\alpha} \circ (j_{\alpha}
\times i_{\alpha}) = i_{\alpha} \circ \lambda_{\alpha}$.
This simply means that the diagram

\[
\begin{CD}
\operatorname{Aut}({\mathbb Q}_{\alpha}) \times
{\mathbb Q}_{\alpha} @>\operatorname{ev}_{\alpha}>>
{\mathbb Q}_{\alpha}\\
@A{j_{\alpha}\times i_{\alpha}}AA @AA{i_{\alpha}}A\\
G_{\alpha}\times X_{\alpha} @>\lambda_{\alpha}>> X_{\alpha}
\end{CD}
\]
\medskip

\noindent commutes for each $\alpha \in B$. Here $\operatorname{ev}_{\alpha}
\colon \operatorname{Aut}({\mathbb Q}_{\alpha}) \times
{\mathbb Q}_{\alpha} \to {\mathbb Q}_{\alpha}$ is the evaluation action, i.e.
$\operatorname{ev}_{\alpha}(g_{\alpha}, x_{\alpha}) = g_{\alpha}(x_{\alpha})$ for each $(g_{\alpha}, x_{\alpha}) \in \operatorname{Aut}({\mathbb Q}_{\alpha}) \times
{\mathbb Q}_{\alpha}$. Let
\[ j  = \times \{ j_{\alpha} \colon \alpha \in B \} \colon
\prod\{ G_{\alpha} \colon \alpha \in B\} \to
\prod\{ \operatorname{Aut}({\mathbb Q}_{\alpha}) \colon \alpha \in B\} \]

\noindent and

\[ i = \times\{ i_{\alpha} \colon \alpha \in B\} \colon
\prod\{ X_{\alpha} \colon \alpha \in B\} \to \prod\{
{\mathbb Q}_{\alpha} \colon \alpha \in B\} .\]

Finally consider the commutative diagram

\[
\begin{CD}
\prod\{ \operatorname{Aut}({\mathbb Q}_{\alpha}) \colon
\alpha \in B\} \times \prod\{
{\mathbb Q}_{\alpha} \colon \alpha \in B\} @>\times\{
\operatorname{ev}_{\alpha} \colon \alpha \in B\}>> \prod\{
{\mathbb Q}_{\alpha} \colon \alpha \in B\}\\
@A{j \times i}AA @AA{i}A\\
\prod\{ G_{\alpha} \colon \alpha \in B\} \times
\prod\{ X_{\alpha} \colon \alpha \in B\} @>\widetilde{\lambda}>>
\prod\{ X_{\alpha} \colon \alpha \in B\}\\
@A{s\times p}AA @AA{p}A\\
G\times X @>\lambda>> X
\end{CD}
\]

\noindent and note that since $|B| = \tau$ the upper horizontal
arrow is actually the action
$\operatorname{ev}_{\tau} \colon \left(\operatorname{Aut}(
{\mathbb Q})\right)^{\tau}\times {\mathbb Q}^{\tau}
\to {\mathbb Q}^{\tau}$. Clearly it suffices to let
$i_{G} = (j \times i) \circ (s \times p)$ and $i_{X} = i \circ p$.
This completes the proof.
\end{proof}

It would be very interesting to prove that for a $\text{AE}(0)$-group
$G$ of weight $\tau > \omega$ the
category of  $AE(0)$-spaces (of weight $\tau$) admitting actions of the
group $G$ and their $G$-maps contains a universal object. For
$\tau = \omega$ this fact has recently been proved in \cite{hjo99}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Closed subgroups of powers of the
symmetric group $S_{\infty}$}\label{SS:symmetric}

The following result gives an embeddability criterion into
the symmetric group $S_{\infty}$ - the group
of all bijections of ${\mathbb N}$ under the relative topology
inherited from ${\mathbb N}^{\mathbb N}$. It is important to note
\cite{dough94} that there exist zero-dimensional Polish groups
which can not be embedded into $S_{\infty}$ as closed subgroups.

\begin{thm}[\cite{bekech96}]\label{T:beke}
Let $G$ be a Polish group. Then the following conditions are equivalent:
\begin{itemize}
\item[(i)]
$G$ is isomorphic to a closed subgroup of $S_{\infty}$;
\item[(ii)]
$G$ admits a (countable) neighborhood basis at the identity
consisting of open subgroups;
\item[(iii)]
$G$ admits a (countable) basis closed under left multiplication
(or a countable basis closed under right multiplication);
\item[(iv)]
$G$ admits a compatible left-invariant ultrametric.
\end{itemize}
\end{thm}

Next we characterize those topological $AE(0)$-groups
which are isomorphic to closed subgroups of infinite powers
$S_{\infty}^{\tau}$, $\tau \geq \omega$, of $S_{\infty}$.

\begin{thm}\label{T:closed}
Let $\tau \geq 1$ be a cardinal number. The following conditions are equivalent for any
topological $AE(0)$-group $G$ of weight $\tau \geq \omega$:
\begin{itemize}
\item[(i)]
$G$ is isomorphic to a closed subgroup of $S_{\infty}^{\tau}$;
\item[(ii)]
$G$ admits a neighborhood basis at the identity
consisting of open subgroups.
\end{itemize}
\end{thm}
\begin{proof}
If $\tau = 1$ our statement coincides with Theorem \ref{T:beke}. Next
consider the case $1 < \tau \leq\omega$. It is easy to see that the group
$S_{\omega}^{\tau}$ admits a countable neighborhood basis
at the identity consisting of open subgroups. Obviously
every closed subgroup of $S^{\omega}_{\tau}$ has the same
property (and, consequently, by Theorem \ref{T:beke}, can be
embedded into $S_{\infty}$). Conversely if a Polish group $G$ admits a countable
neighborhood basis at the identity
consisting of open subgroups, then, by Theorem \ref{T:beke},
$G$ is isomorphic to a closed subgroup of $S_{\infty}$. It only
remains to note that $G_{\infty}$ is isomorphic to a closed
subgroup of $S_{\infty}^{\tau}$ for any $\tau$.

Next we assume that $\tau > \omega$. By
\cite[Lemma 8.2.1]{chibook96}, $G$ is isomorphic
to the limit space of a factorizing $\omega$-spectrum
${\mathcal S}_{G} = \{ G_{\alpha}, p_{\alpha}^{\beta}, A\}$
all spaces $G_{\alpha}$, $\alpha \in A$, of which are Polish
groups and all limit projections $p_{\alpha} \colon G \to G_{\alpha}$,
$\alpha \in A$,
of which are $0$-soft homomorphisms.

Now consider the following relation $L \in A^{2}$:
\begin{multline*}
 L = \left\{ (\alpha ,\beta ) \in A^{2} \colon \alpha \leq \beta
\;\text{and there exists a countable neighborhood basis}\right.\\
 \;\;\;\;\;\;\;\; {\mathcal V}_{\alpha} \;\text{at
the identity}\; e_{\alpha} \;\text{of}\; G_{\alpha} \; \text{containing
intersections of its finite subcoll-}\\
\;\;\;\;\;\;\;\;\text{ections and such that for each}\; V \in {\mathcal V}_{\alpha}\;\;
\text{there is an open subgroup} \; U^{\beta ,\alpha}_{V}\\
\left.  \; \text{of}\; G_{\beta}\;
\text{with}\; U_{V}^{\beta ,\alpha} \subseteq
\left( p_{\alpha}^{\beta}\right)^{-1}(V)\; \right\}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;
\end{multline*}

Let us verify conditions of Proposition \ref{P:search}.

{\em Existence}.
For each $\alpha \in A$ we need to find
$\beta \in A$  such that  $(\alpha ,\beta ) \in L$.
Let ${\mathcal V}_{\alpha}$ be a countable
neighborhood basis at $e_{\alpha} \in G_{\alpha}$ which contains
intersections of its finite subcollections. For each $V \in {\mathcal V}_{\alpha}$
the set $p_{\alpha}^{-1}\left( V\right)$ is a
neighborhood of the identity $e \in G$. By (ii), there exists an
open subgroup $U_{V}$ of $G$ such that
$U_{V} \subseteq p_{\alpha}^{-1}\left( V\right)$.
Since every open subgroup in $G$ is closed, it follows that $U_{V}$, as
an open and closed subset of $G$, is a functionally open in $G$.
Recall that the spectrum
${\mathcal S}_{G}$
is factorizing and consequently there exist an index
$\beta_{V} \in A$ and an
open subset $U_{V}^{\beta_{V}} \subseteq G_{\beta_{V}}$ such
that $\beta_{V} \geq \alpha$
and $U_{V} = p_{\beta_{V}}^{-1}\left( U_{V}^{\beta_{V}}\right)$.
By Corollary \ref{C:3.1.2}, there exists an index $\beta \in A$ such that
$\beta \geq \beta_{V}$ for each $V \in {\mathcal V}_{\alpha}$.
Let $U_{V}^{\beta ,\alpha} = \left( p_{\beta_{V}}^{\beta}\right)^{-1}\left( U_{V}^{\beta_{V}}\right)$,
$V \in {\mathcal V}_{\alpha}$. Note that

\begin{multline*}
U_{V}^{\beta ,\alpha} =
\left( p_{\beta_{V}}^{\beta}\right)^{-1}\left( U_{V}^{\beta_{V}}\right)
= p_{\beta}\left( p_{\beta_{V}}\left( U_{V}^{\beta_{V}}\right)\right)
= p_{\beta}(U_{V})
\subseteq p_{\beta}\left( p_{\alpha}^{-1}(V)\right) =\\
 \left( p_{\alpha}^{\beta}\right)^{-1}(V)
\;\text{for each}\; V \in {\mathcal V}_{\alpha} .
\end{multline*}

\noindent Since the limit projection $p_{\beta} \colon G \to G_{\beta}$ is a
homomorphism it follows that $U_{V}^{\beta ,\alpha} = p_{\beta}(U_{V})$
is a subgroup of $G_{\beta}$. Finally since $U_{V}^{\beta_{V}}$
is open in $G_{\beta_{V}}$, we conclude that $U_{V}^{\beta ,\alpha} = \left( p_{\beta_{V}}^{\beta}\right)^{-1}\left( U_{V}^{\beta_{n}}\right)$
is open in $G_{\beta}$. This shows that $(\alpha ,\beta ) \in L$.

{\em Majorantness}.
If  $(\alpha ,\beta ) \in L$  and
$\gamma \geq \beta$, then  $(\alpha ,\gamma ) \in L$.
Since $(\alpha ,\beta ) \in L$ it follows that for each
$V \in {\mathcal V}_{\alpha}$ there exists an open subgroup $U_{V}^{\beta ,\alpha}$
of $G_{\beta}$ such that
$U_{V}^{\beta ,\alpha} \subseteq \left( p_{\alpha}^{\beta}\right)^{-1}
(V)$ where
${\mathcal V}_{\alpha}$ is a countable
neighborhood basis at the identity $e_{\alpha} \in G_{\alpha}$
containing intersections of its finite subcollections.
Let $U_{V}^{\gamma ,\alpha} =
\left( p_{\beta}^{\gamma}\right)^{-1}(U_{V}^{\beta ,\alpha})$ for
each $V \in {\mathcal V}_{\alpha}$. Since the projection
$p_{\beta}^{\gamma} \colon G_{\gamma} \to G_{\beta}$ is a
continuous homomorphism, it follows that $U_{V}^{\gamma ,\alpha}$ is an
open subgroup of $G_{\gamma}$. Obviously

\[ U_{V}^{\gamma ,\alpha} = \left( p_{\beta}^{\gamma}\right)^{-1}(U_{V}^{\beta ,\alpha})
\subseteq  \left( p_{\beta}^{\gamma}\right)^{-1}\left(\left(
p_{\alpha}^{\beta}\right)^{-1}(V)\right) =
\left( p_{\alpha}^{\gamma}\right)^{-1}(V) ,\;\; V \in {\mathcal V}_{\alpha} , \]

\noindent which shows that $(\alpha ,\gamma ) \in L$ as required.

{\em $\omega$-closeness}.
Let $\{ \alpha_{i} : i \in \omega \}$
be a countable chain in $A$ and
$(\alpha_{i}, \beta ) \in L$ for some
$\beta \in A$ and each $i \in \omega$. We need to show that
$(\alpha ,\beta ) \in L$ where $\alpha =
\sup \{\alpha_{i} \colon i \in \omega \}$.

Let ${\mathcal V}_{\alpha_{i}}$ be a countable
neighborhood basis
at $e_{\alpha_{i}} \in G_{\alpha_{i}}$ and
$U_{V}^{\beta ,\alpha_{i}}$, $V \in {\mathcal V}_{\alpha_{i}}$,
be an open subgroup of $G_{\beta}$ witnessing the fact that
$(\alpha_{i}, \beta ) \in L$.

Consider the collection
\[ {\mathcal V}_{\alpha} =
\bigcup\{ \left( p_{\alpha_{i}}^{\alpha}\right)^{-1}\left(
{\mathcal V}_{\alpha_{i}}\right)
\colon i \in \omega\}  .\]

\noindent Since the spectrum ${\mathcal S}_{G}$ is an $\omega$-spectrum and
since $\alpha = \sup\{ \alpha_{i} \colon i \in \omega\}$ it
follows that ${\mathcal V}_{\alpha}$ forms a neighborhood basis
at $e_{\alpha} \in G_{\alpha}$. For each $\widetilde{V} \in {\mathcal V}_{\alpha}$
choose $V \in {\mathcal V}_{\alpha_{i}}$ such that $\widetilde{V} = \left( p_{\alpha_{i}}^{\alpha}\right)^{-1}(V)$ and let $U_{\widetilde{V}}^{\beta ,\alpha} = U_{V}^{\beta ,\alpha_{i}}$. Since $(\alpha_{i} ,\beta) \in L$, we have

\[ U_{\widetilde{V}}^{\beta ,\alpha} = U_{V}^{\beta ,\alpha_{i}} \subseteq \left( p_{\alpha_{i}}^{\beta}\right)^{-1}(V) = \left( p_{\alpha}^{\beta}\right)^{-1}
\left( \left( p_{\alpha_{i}}^{\alpha}\right)^{-1}(V)\right) = \left( p_{\alpha}^{\beta}\right)^{-1}(\widetilde{V}) .\]

\noindent This proves that $(\alpha ,\beta ) \in L$.

According to Proposition \ref{P:search} the set $\widetilde{A}$ of
$L$-reflexive elements is cofinal and $\omega$-closed in $A$. The
$L$-reflexivity of an element $\alpha \in A$ means precisely
that there exists a
countable neighborhood basis ${\mathcal V}_{\alpha}$ at
$e_{\alpha} \in G_{\alpha}$ containing intersections of its finite
subcollections and such that for each $V \in {\mathcal V}_{\alpha}$
there exists an open subgroup $U_{V}^{\alpha} \subseteq G_{\alpha}$
with $U_{V}^{\alpha} \subseteq V$. This obviously means that for each
$\alpha \in {\widetilde{A}}$ the Polish group $G_{\alpha}$
satisfies condition (ii)
of Theorem \ref{T:beke}. Consequently, by Theorem \ref{T:beke},
$G_{\alpha}$ is topologically isomorphic to a closed subgroup
of $S_{\infty}$.
Next note that since $\widetilde{A}$ is cofinal in $A$ the limit
space of the spectrum
${\mathcal S} = \{ G_{\alpha}, p_{\alpha}^{\beta}, \widetilde{A}\}$
is topologically isomorphic to $G$. This obviously implies that $G$
is isomorphic to a closed subgroup of the product
$\displaystyle \prod\{ G_{\alpha} \colon \alpha \in \widetilde{A}\}$,
which in turn is topologically isomorphic to a closed
subgroup of $S_{\infty}^{\tau}$ (note that
$|\widetilde{A}| = w(G) = \tau$). This completes
the proof of implication (ii) $\Longrightarrow$ (i).

Verification of the implication (i) $\Longrightarrow$ (ii)
is trivial. Proof is completed.
\end{proof}

\begin{cor}\label{C:polish}
Let $\tau \geq 2$. The following conditions are equivalent
for any Polish group $G$:
\begin{itemize}
\item[(i)]
$G$ is topologically isomorphic to a closed subgroup of $S_{\infty}$;
\item[(ii)]
$G$ is topologically isomorphic to a closed subgroup of $S_{\infty}^{\tau}$.
\end{itemize}
\end{cor}

\begin{cor}\label{C:doug}
There exists a zero-dimensional Polish group which is not
topologically isomorphic to a closed subgroup of
$S_{\infty}^{\tau}$ for any cardinal number $\tau$.
\end{cor}
\begin{proof}
It is known \cite{dough94} that there exists a zero-dimensional
Polish group $G$ which is not topologically isomorphic to a
closed subgroup of $S_{\infty}$. By Corollary \ref{C:doug}, $G$
can not be topologically isomorphic to a closed subgroup of
$S_{\infty}^{\tau}$ for
any cardinal $\tau \geq 2$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Baire isomorphisms}\label{SS:baire}
The main result of this Subsection (Theorem \ref{T:baire})
allows us to reduce in many instances (descriptive) set
theoretical considerations of general $\text{AE}(0)$-groups
to those for Polish groups.

\begin{lem}\label{L:dixmier}
Let $f \colon X \to Y$ be a $0$-soft homomorphism between
$\text{AE}(0)$-groups. Then there
exists a Baire isomorphism $h \colon Y \times \ker f \to X$ such
that $f \circ h = \pi_{Y}$, where
$\pi_{Y} \colon Y \times \ker f \to Y$ stands for the projection
onto the first coordinate.
\end{lem}
\begin{proof}
By Proposition \ref{P:bb}, there exists a Baire map
$g \colon Y \to X$ such that
$f \circ g = \operatorname{id}_{Y}$.
The required Baire isomorphism $h \colon Y \times \ker f \to X$
(not a homomorphism unless $g$
is a homomorphism itself) can now be defined by letting
\[ h(y,a) = g(y)\cdot a ,\;\;\text{for each}\;\; (y,a) \in Y \times \ker f ,\]

\noindent where $\cdot$ denotes the multiplication operation in $X$.
\end{proof}

\begin{thm}\label{T:baire}
Every $\text{AE}(0)$-group is Baire isomorphic to the product of Polish groups.
\end{thm}
\begin{proof}
Let $X$ be a $\text{AE}(0)$-group. If $w(X) = \omega$, then $X$ itself is Polish
and there is nothing to prove.

Let now $w(X) = \tau > \omega$. According to Theorem \ref{T:ae0},
$X$ is topologically and algebraically isomorphic to the limit of a
well-ordered continuous spectrum
${\mathcal S}_{X} = \{ X_{\alpha}, p_{\alpha}^{\alpha +1}, \tau \}$
such that $X_{0}$ is a Polish group and the $0$-soft homomorphism
$p_{\alpha}^{\alpha +1} \colon X_{\alpha +1} \to X_{\alpha}$ has a
Polish kernel for each $\alpha < \tau$.

Our goal is to prove that $X$ is Baire isomorphic to the product
$\displaystyle X_{0} \times \prod\{
\ker p_{\alpha}^{\alpha +1} \colon \alpha < \tau\}$. We proceed by induction.
By Lemma \ref{L:dixmier}, there exists a Baire isomorphism
$h_{1} \colon X_{0} \times \ker p_{0}^{1} \to X_{1}$ such
that $p_{0}^{1} \circ h_{1} = \pi_{X_{0}}$.
Suppose that for each $\alpha$, where $1 \leq \alpha < \beta < \tau$, we
have already constructed Baire isomorphism
$\displaystyle h_{\alpha} \colon X_{0} \times \prod\{
\ker p_{\delta}^{\delta +1}\colon \delta < \alpha\} \to X_{\alpha}$
in such a way that
\begin{enumerate}
\item
If $\alpha + 1 < \beta$, then
\[
\begin{CD}
X_{0} \times \prod\{ \ker p_{\delta}^{\delta +1}\colon \delta <
\alpha +1\} @>h_{\alpha +1}>> X_{\alpha +1}\\
@V{\operatorname{id}_{X_{0}}\times \pi_{\alpha}^{\alpha +1}}VV @VV
{p_{\alpha}^{\alpha +1}}V\\
X_{0}\times \prod\{ \ker p_{\delta}^{\delta +1}\colon
\delta < \alpha\} @>h_{\alpha}>> X_{\alpha},
\end{CD}
\]

\noindent where
\begin{multline*}
 \pi_{\alpha}^{\alpha +1} \colon \prod\{ \ker
p_{\delta}^{\delta +1}\colon \delta < \alpha +1\}  =\\
\prod\{ \ker p_{\delta}^{\delta +1}\colon \delta < \alpha\}
\times X_{\alpha} \to \prod\{ \ker
p_{\delta}^{\delta +1}\colon \delta < \alpha\}
\end{multline*}

\noindent is the natural projection.
\item
$h_{\alpha} = \lim\{ h_{\gamma} \colon \gamma < \alpha\}$, whenever
$\alpha <\beta$ is a limit ordinal number.
\end{enumerate}

We now construct Baire isomorphism
$\displaystyle h_{\beta} \colon X_{0} \times \prod\{
\ker p_{\delta}^{\delta +1}\colon \delta < \beta\} \to X_{\beta}$

If $\beta$ is a limit ordinal number, then we let
$h_{\beta} = \lim\{ h_{\alpha} \colon \alpha < \beta\}$.

If $\beta = \alpha +1$, then consider the following commutative diagram

\[
\begin{CD}
\left( X_{0} \times \prod\{ \ker p_{\delta}^{\delta +1}\colon \delta <
\alpha\}\right) \times \ker p_{\alpha}^{\alpha +1} @>h_{\alpha}\times \operatorname{id}>> X_{\alpha} \times \ker p_{\alpha}^{\alpha +1} @>h>> X_{\alpha +1}\\
@V{\pi_{1}}VV   @V{\pi_{X_{\alpha}}}VV @VV
{p_{\alpha}^{\alpha +1}}V\\
X_{0}\times \prod\{ \ker p_{\delta}^{\delta +1}\colon
\delta < \alpha\} @>h_{\alpha}>> X_{\alpha}
@>\operatorname{id}_{X_{\alpha}}>> X_{\alpha},
\end{CD}
\]
\bigskip

\noindent where

\begin{itemize}
\item[(a)]
$\operatorname{id} \colon \ker p_{\alpha}^{\alpha +1} \to
\ker p_{\alpha}^{\alpha +1}$ stands for the identity map;
\item[(b)]
\[ \pi_{1} \colon \left( X_{0} \times \prod\{ \ker
p_{\delta}^{\delta +1}\colon \delta <
\alpha\}\right) \times \ker p_{\alpha}^{\alpha +1}
\to X_{0} \times \prod\{ \ker p_{\delta}^{\delta +1}\colon \delta <
\alpha\}\]

\noindent denotes the projection onto the first coordinate and
\item[(c)]
$h \colon X_{\alpha} \times \ker p_{\alpha}^{\alpha +1} \to
X_{\alpha +1}$ is a Baire isomorphism
existence of which is guaranteed by Lemma \ref{L:dixmier}.
\end{itemize}
The required Baire isomorphism $\displaystyle h_{\alpha +1}
\colon X_{0} \times \prod\{ \ker p_{\delta}^{\delta +1} \colon
\delta < \alpha +1\} \to X_{\alpha +1}$ can now be defined as
the composition $h_{\alpha +1} = h\circ \left( h_{\alpha} \times
\operatorname{id}\right)$.
This completes induction and finishes the construction of Baire
isomorphisms
$h_{\alpha}$, $\alpha < \tau$. It is now easy to see that
\[ h = \lim\{ h_{\alpha} \colon \alpha < \tau\} \colon
X_{0} \times \prod\{ \ker p_{\alpha}^{\alpha +1} \colon \alpha < \tau \} \to X\]

\noindent is the required Baire isomorphism. Note here that $X_{0}$ as well as
$\ker p_{\alpha}^{\alpha +1}$, $\alpha < \tau$, are Polish groups.
Proof is completed.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}

\bibitem{bekech96}
H.~Becker and A.~Kechris, \emph{The {D}escriptive {S}et {T}heory of {P}olish
  {G}roup {A}ctions}, London Math. Soc. Lecture Note Series, no. 232, Cambridge
  University Press, Cambridge, 1996.

\bibitem{bellchi93}
M.~Bell and A.~Chigogidze, \emph{Topological groups homeomorphic to products of
  discrete spaces}, Topology Appl. \textbf{53} (1993), 67--73.

\bibitem{chi994}
A.~Chigogidze, \emph{On spaces {B}aire isomorphic to the powers of the real
  line}, Topology Appl., to appear.

\bibitem{chibook96}
\bysame, \emph{Inverse {S}pectra}, North Holland, Amsterdam, 1996.

\bibitem{chi972}
\bysame, \emph{Compact groups and absolute extensors}, Topology Proceedings
  \textbf{22} (1997), 63--70.

\bibitem{chi993}
\bysame, \emph{Topological characterization of torus groups}, Topology Appl.
  \textbf{94} (1999), no.~1, 13--25.

\bibitem{dough94}
R.~Dougherty, \emph{Examples of non-shy sets}, Fund. Math. \textbf{144} (1994),
  73--88.

\bibitem{hay74}
R.~Haydon, \emph{On a problem of {P}elczynski: {M}ilutin spaces, {D}ugundji
  spaces and $\text{AE}(0-\dim )$}, Studia Math. \textbf{52} (1974), 23--31.

\bibitem{hjo99}
G.~Hjorth, \emph{A universal {P}olish $\text{G}$-space}, Topology Appl.
  \textbf{91} (1999), 141--150.

\bibitem{hofmor98}
K.~H. Hofmann and S.~A. Morris, \emph{The {S}tructure of {C}ompact {G}roups},
  de {G}ruyter {S}tudies in {M}athematics, no.~25, de {G}ruyter, Berlin, 1998.

\bibitem{kech95}
A.~Kechris, \emph{Classical {D}escriptive {S}et {T}heory}, Springer-Verlag, New
  York, 1995.

\bibitem{megr96}
M.~Megrelishvili, \emph{Free topological $\text{G}$-spaces}, New Zealand Journ.
  Mathematics \textbf{125} (1996), no.~1, 59--72.

\bibitem{pest99}
V.~Pestov, \emph{Topological groups: where to from here?},  (1999), preprint.

\bibitem{pont46}
L.~S. Pontryagin, \emph{Topological {G}roups}, Princeton University Press,
  Princeton, New Jersey, 1946.

\bibitem{usp86}
V.~V. Uspenskii, \emph{A universal topological group with a countable base},
  Func. Anal. and its Appl. \textbf{20} (1986), 160--161.

\end{thebibliography}

%%\bibliographystyle{amsplain}
%%\bibliography{mainbiblio}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Signal processing strategy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Two signal processing strategies are applied: newly proposed image processing based on the FCN and conventional one which is the adaptive wavenumber filtering.
The former one is represented by the left branch on the scheme shown in~Fig.~\ref{fig:sig_proc_strategy} whereas the latter one is represented by the right branch.
The starting point for both approaches is the dataset consisted of frames of propagating guided waves.
However, the FCN takes as an input the RMS which combines all frames into one image.
It essentially represents the spatial energy distribution of propagating waves.
In the adaptive wavenumber filtering method, all frames are utilised (right branch of Fig.~\ref{fig:sig_proc_strategy}).
The RMS is applied later on to output final damage map in the form of an image.

It should be noted that we tested two types of activation functions as a last layer of the FCN: softmax and sigmoid.
The former one gives the binary output which segments pixels in two categories: damaged and undamaged.
On the other hand, the threshold needs to be applied when the sigmoid activation function is used.
Also, damage maps resulted from the adaptive wavenumber filtering must be thresholded so that the area and shape of delamination can be estimated.

Finally, IoU is applied as a measure of delamination size, shape and location so that all presented approaches can be compared quantitatively.
	\begin{figure}
		\centering
		\includegraphics[scale=0.8]{fig5.png}
		\caption{Diagram of signal processing strategy by the proposed Fully Convolutional Network (left branch) in comparison to signal processing utilising adaptive wavenumber filtering method (right branch). }
		\label{fig:sig_proc_strategy}
	\end{figure}
The particular blocks presented in the Fig.~\ref{fig:sig_proc_strategy} will be explained in more details in the next sections.


\documentclass[letterpaper, 12pt]{article}

%% Language and font encodings
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}

%% Sets page size and margins
\usepackage[letterpaper,top=2.5cm,bottom=2cm,left=2cm,right=2cm,marginparwidth=1.75cm]{geometry}

%% Useful packages
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{physics}
\usepackage{bbold}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}
\usepackage{listings}
\usepackage{multicol}
\usepackage{float}
\usepackage{enumitem}

\usepackage{bm}
\date{\today}

\title{CSC411 Assignment 3}
\author{Yue Guo}
\begin{document}
\maketitle

%%%%%%%%%%%Q1 	20 NEWS GROUP  %%%%%%%%%%%%%%
\section{20 News Group}
\subsection{Top 3 Algorithms}
Note: I have added a list of the names of all 20 categories in in load\_data( ) function for debugging purposes. This is does not affect my classification algorithm.

\subsubsection{Neural Network}
\begin{enumerate}

    \item Hypterparameter: number of layers

	Method: K-fold cross validation

	II have tried a single layer neural network vs multi-layered neural network.  It turns out that the single neural network is the fastest and also most accurate.

	\item Train/test loss
	\begin{itemize}

     \item  Train accuracy 0.974721583878
     \item Test accuracy 0.663303239511
        \end{itemize}

      \item Reason to pick this algorithm

      Neural network is commonly used in NLP, according to our lecture on neural networks. It is good at handling large data set with any features and produce a non-linear decision boundary.

      \item My expectations

      This meets my expectation because NNs are good at working with a large dataset with many feartures. When I increase the number of hidden layers, accuracy decreases. I think this is because it overfits.

\end{enumerate}

\subsubsection{Random forest}
\begin{enumerate}

    \item Hypterparameter: number of estimators

	Method: K-fold cross validation

	I have tried Ensamble with 10 to 150 estimators, and 150 works best after comparing results from cross validation.

	\item Train/test loss
	\begin{itemize}
     \item  Train accuracy 0.974721583878
     \item Test accuracy 0.59346787042
        \end{itemize}

      \item Reason to pick Random forest
      I have tried decision tree, with different parameters(entropy vs gini) and reduced features, but all end up overfitting. Random forest introduces more randomness and also scales the weights of misclassified data in each iteration. To see decision tree, please uncomment from line 272 to 290 in q1.py.

      \item My expectations

      This meets my expectations because Random Forest adds more randomness in each step, and ensamble method is more resistant to overfitting because it assigns weights to different features in each step. I also tried decision\_tree, and it does not generalize well.
      The number of estimators learns better with a larger group of weak learners.

\end{enumerate}

\subsubsection{SVM - Best Classifier}
\begin{enumerate}

    \item Hypterparameter: rand\_state

	Method: K-fold cross validation

	In the code, I have tried different rand\_state and cross validated each. It turns out $rand\_state = 0$ is the best

	\item Train/test loss
	\begin{itemize}
     \item  Train accuracy  0.972511932119
     \item Test accuracy 0.691980881572
        \end{itemize}

 	\item Reason to pick SVM
	Because of the natural of this given problem, it is still a classification problem. Using neural nets might be an overkill. SVM produces a linear decision boundary with a margin for two classes. It can be extended to multi-class using algorithms such as one-vs-all.

          \item My expectations

      SVM is the best out of all. I think it is because it has a linear decision boundary with a margin and does not overfit on training data. The test accuracy is close to single neuron neural net, but still higher.

        \item Test confusion matrix

        The most confused classes are class 16 talk.politics.guns and 18 talk.politics.misc

\resizebox{\columnwidth}{!}{%
\setcounter{MaxMatrixCols}{20}
$\begin{matrix}
157.0 & 6.0 & 4.0 & 0.0 & 2.0 & 0.0 & 1.0 & 6.0 & 2.0 & 3.0 & 1.0 & 4.0 & 3.0 & 7.0 & 6.0 & 19.0 & 5.0 & 22.0 & 12.0 & 31.0 \\
2.0 & 278.0 & 19.0 & 12.0 & 3.0 & 47.0 & 3.0 & 1.0 & 3.0 & 2.0 & 2.0 & 7.0 & 9.0 & 9.0 & 10.0 & 2.0 & 2.0 & 1.0 & 1.0 & 4.0 \\
3.0 & 18.0 & 242.0 & 37.0 & 10.0 & 36.0 & 6.0 & 4.0 & 4.0 & 0.0 & 1.0 & 7.0 & 12.0 & 2.0 & 3.0 & 2.0 & 5.0 & 1.0 & 1.0 & 1.0 \\
2.0 & 8.0 & 37.0 & 254.0 & 35.0 & 9.0 & 14.0 & 3.0 & 1.0 & 4.0 & 0.0 & 3.0 & 26.0 & 1.0 & 2.0 & 1.0 & 2.0 & 4.0 & 2.0 & 2.0 \\
1.0 & 6.0 & 18.0 & 22.0 & 267.0 & 4.0 & 12.0 & 2.0 & 3.0 & 1.0 & 2.0 & 4.0 & 10.0 & 2.0 & 3.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 \\
0.0 & 24.0 & 14.0 & 9.0 & 4.0 & 273.0 & 0.0 & 2.0 & 0.0 & 1.0 & 0.0 & 4.0 & 8.0 & 1.0 & 2.0 & 1.0 & 0.0 & 0.0 & 0.0 & 1.0 \\
2.0 & 5.0 & 3.0 & 13.0 & 9.0 & 2.0 & 311.0 & 12.0 & 4.0 & 7.0 & 1.0 & 6.0 & 16.0 & 3.0 & 3.0 & 1.0 & 2.0 & 0.0 & 0.0 & 2.0 \\
7.0 & 3.0 & 3.0 & 2.0 & 6.0 & 0.0 & 6.0 & 282.0 & 21.0 & 3.0 & 3.0 & 3.0 & 9.0 & 6.0 & 8.0 & 2.0 & 7.0 & 1.0 & 6.0 & 3.0 \\
6.0 & 3.0 & 2.0 & 0.0 & 5.0 & 1.0 & 6.0 & 15.0 & 302.0 & 6.0 & 2.0 & 3.0 & 10.0 & 4.0 & 5.0 & 1.0 & 8.0 & 5.0 & 3.0 & 1.0 \\
12.0 & 10.0 & 18.0 & 8.0 & 14.0 & 6.0 & 11.0 & 28.0 & 16.0 & 329.0 & 23.0 & 19.0 & 14.0 & 15.0 & 18.0 & 15.0 & 13.0 & 12.0 & 9.0 & 9.0 \\
1.0 & 0.0 & 2.0 & 1.0 & 1.0 & 0.0 & 2.0 & 2.0 & 1.0 & 20.0 & 345.0 & 3.0 & 2.0 & 4.0 & 3.0 & 0.0 & 0.0 & 1.0 & 4.0 & 3.0 \\
2.0 & 6.0 & 2.0 & 2.0 & 4.0 & 3.0 & 1.0 & 1.0 & 0.0 & 1.0 & 2.0 & 282.0 & 12.0 & 1.0 & 1.0 & 0.0 & 8.0 & 4.0 & 3.0 & 2.0 \\
8.0 & 8.0 & 2.0 & 27.0 & 15.0 & 4.0 & 7.0 & 13.0 & 8.0 & 1.0 & 1.0 & 10.0 & 230.0 & 8.0 & 14.0 & 2.0 & 1.0 & 3.0 & 2.0 & 1.0 \\
8.0 & 0.0 & 4.0 & 0.0 & 0.0 & 2.0 & 0.0 & 3.0 & 7.0 & 4.0 & 3.0 & 2.0 & 12.0 & 302.0 & 7.0 & 7.0 & 6.0 & 2.0 & 5.0 & 7.0 \\
11.0 & 8.0 & 9.0 & 2.0 & 4.0 & 5.0 & 0.0 & 6.0 & 7.0 & 1.0 & 3.0 & 4.0 & 9.0 & 4.0 & 289.0 & 2.0 & 8.0 & 2.0 & 11.0 & 6.0 \\
44.0 & 1.0 & 1.0 & 0.0 & 2.0 & 1.0 & 2.0 & 1.0 & 5.0 & 6.0 & 2.0 & 4.0 & 2.0 & 7.0 & 5.0 & 319.0 & 11.0 & 11.0 & 3.0 & 63.0 \\
6.0 & 2.0 & 2.0 & 0.0 & 3.0 & 0.0 & 4.0 & 3.0 & 4.0 & 1.0 & 4.0 & 12.0 & 3.0 & 4.0 & 6.0 & 0.0 & 243.0 & 6.0 & 86.0 & 20.0 \\
13.0 & 2.0 & 3.0 & 0.0 & 0.0 & 0.0 & 1.0 & 5.0 & 2.0 & 0.0 & 1.0 & 3.0 & 2.0 & 6.0 & 2.0 & 1.0 & 9.0 & 283.0 & 7.0 & 6.0 \\
8.0 & 1.0 & 6.0 & 3.0 & 1.0 & 0.0 & 2.0 & 6.0 & 6.0 & 7.0 & 0.0 & 12.0 & 2.0 & 7.0 & 7.0 & 3.0 & 21.0 & 15.0 & 145.0 & 10.0 \\
26.0 & 0.0 & 3.0 & 0.0 & 0.0 & 2.0 & 1.0 & 1.0 & 2.0 & 0.0 & 3.0 & 4.0 & 2.0 & 3.0 & 0.0 & 20.0 & 12.0 & 3.0 & 10.0 & 79.0 \\
\end{matrix}%
$}

    \end{enumerate}

\subsubsection{Bernoulli Baseline}
\begin{enumerate}
	\item Train/test loss
	\begin{itemize}
     \item  Train accuracy  0.598727240587
     \item Test accuracy 0.457912904939
        \end{itemize}
\end{enumerate}

%%%%%%%%%%%Q2 SVM %%%%%%%%%%%%%%
\section{SVM}
\subsection{SVM test}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{q2part1plot.png}
\caption{\label{}Plot of test SVM}
\end{figure}

\subsection{SVM code}
see code

\subsection{SVM on MINIST}
\subsubsection{without momentum}
\begin{enumerate}
	\item Train loss$=0.400699029921$
	\item Test loss$=0.37243523202$
	\item classification accuracy on training set $= 0.826985854189$
	\item classification accuracy on testing set $=0.818503401361$
	\item Plot of w
	\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{q2_3_1.png}
\caption{\label{}Plot momentum = 0 }
\end{figure}
\end{enumerate}

\subsubsection{with momentum}
\begin{enumerate}
	\item Train loss = 0.424045274211
	\item Test loss = 0.394698671058
	\item classification accuracy on training set $= 0.817555313747$
	\item classification accuracy on testing set $=0.809977324263$
	\item Plot of w
	\begin{figure}[H]
	\centering
	\includegraphics[width=0.5\textwidth]{q_2_3_2.png}
	\caption{\label{}Plot momentum = 0.1 }
	\end{figure}
\end{enumerate}

%%%%%%%%%%%Q3%%%%%%%%%%%%%%
\section{Kernels}
%%%%%%%%%% q 3.1%%%%%%%%%%%%%%
\subsection{Positive semidefinite and quadratic form}
Assume K is symmetric, we can decompose K into $U \Lambda U^T$
\begin{equation*}
\begin{split}
x^T K x &= x^T (U \Lambda U^T) x = (x^T U) \Lambda (U^T x)\\
\end{split}
\end{equation*}

$\Lambda$ has the eigenvalues $\lambda_{i}$, and if $K$ is positive, and all $\lambda_{i}$ > 0,

\begin{equation*}
\begin{split}
x^T K x &= \Sigma_{i =1}^{d} \lambda_{i} ([x^T U_{i}])^2 >= 0\\
\end{split}
\end{equation*}

Then $x^T K x >= 0$ for all $x$ in $ \mathbb{R}^{d}$ iff $K$ is postive semidefinite

%%%%%%%%%Q 3.2 %%%%%%%%%%%%%%
\subsection{Kernel properties}
%%%%%%%% Q 3.2 q2%%%%%%%%%%%%%
\subsubsection{$\alpha$}
Define mapping $\phi (x) = \sqrt{\alpha}$, $\alpha > 0$, and the kernel $\langle \phi(x), \phi(y) \rangle = \alpha$.
The resulting matrix K has item $K_{ij} = \alpha $, the matrix K has equal number of row and columns, and element is $\alpha$. Since $\alpha$ > 0, and all elements are equal, K is positive semidefinite

 %%%%%%%Q 3.2 q3 %%%%%%%%%%%%%%
\subsubsection{$f(x), f(y)$}
$K_{ij} = \langle \phi (x),  \phi (y) \rangle$, \\
define $\phi(x) = f(x), \forall f: \mathbb{R}^{d} \rightarrow \mathbb{R}$\\
define $\phi(y) = f(y), \forall f: \mathbb{R}^{d} \rightarrow \mathbb{R}$\\
Since f(x) and f(y) produce a scalar,  $\langle \phi (x),  \phi (y) \rangle = f(x) \cdot f(y)$

%%%%%%%%Q3.2 part 3%%%%%%%%%%%
\subsubsection{k1 and k2}
If the gram matrix, $K_{1}$ of kernel k1 and gram matrix, $K_{2}$ of kernel k2 are positive semidefinite, by scaling them and adding each element, the new gram matrix of $a \cdot k_{1}(x, y) + b \cdot k_{2}(x, y)$, call it $K$, each element of K is positive since a ,b > 0.\\
$K$ is also symmetric because $K_{1}$ and $K_{2}$ are symmetric with the same dimension, and element wise addition and linear combination preserve the symmetric property.\\

%%%%%%%%Q3.2 part 4%%%%%%%%%%%
\subsubsection{$k(x, y) = \frac{k_{1}(x ,y) }{\sqrt{k_{1}(x, x)} \sqrt{k_{1}(y, y)} }$}
Let $\phi_{1}$ be the mapping defined by $k_{1}$\\
We define a new mapping, $\phi$ for $k(x ,y)$\\
We let $\phi(x) = \frac{\phi_{1} (x)}{\norm{\phi_{1}(x)}}$\\
\begin{equation*}
\begin{split}
k(x, y) &= \langle \phi (x), \phi (y) \rangle \\
&= \frac{\phi_{1}(x)}{\norm{\phi_{1}(x)}} \cdot  \frac{\phi_{1}(y)}{\norm{\phi_{1}(y)}}\\
&= \frac{\phi_{1}(x)}{\sqrt{\phi_{1}(x) \cdot \phi_{1}(x)}} \cdot \frac{\phi_{1}(y)}{\sqrt{\phi_{1}(y) \cdot \phi_{1}(y)}}\\
&= \frac{\phi_{1}(x)}{( \sqrt{\phi_{1}(x)} \cdot \sqrt{ \phi_{1}(y)}  )}  \cdot \frac{\phi_{1}(y)}{( \sqrt{\phi_{1}(x)} \cdot \sqrt{ \phi_{1}(y)}  )} \\
&= \frac{\phi_{1}(x)}{ \sqrt{\phi_{1}(x) \cdot  \phi_{1}(y) }}  \cdot  \frac{\phi_{1}(x)}{ \sqrt{\phi_{1}(x) \cdot  \phi_{1}(y) }}\\
 k(x, y) &= \frac{k_{1}(x ,y) }{\sqrt{k_{1}(x, x)} \sqrt{k_{1}(y, y)} }
\end{split}
\end{equation*}
Therefore, there is a new mapping $\phi(x)$ that supports $k(x, y)$ and it is a kernel because $\phi(x)$
is the product of two kernel mappings

\end{document}


\chapter{Market Basics}
\label{ch:marketbasics}

\newthought{This section deals with some basic stock market information and a brief introduction into fundamental stock analysis.}

\section{Description of Basic Securities}

\textbf{Bonds}\sidenote{sometimes called debt} are typically based on debt - that is bond holders lend their money to the bond issuer in return for a ``fixed income''. The fixed income is in the form of periodic interest payments on the amount loaned. At the end of the bond term the issuer repays the original amount. Bonds are considered low risk, where the issuer is a stable entity such as a government with a strong underlying economy. As a result of the low risk the return is also low.

Bonds can be traded on the secondary market, and are usually sold at a market price which is not the same as the face value of the bond.\sidenote{Face Value is the amount to be repaid at maturity} Bonds can be ``fixed coupon'' --- which pays a fixed (percentage) amount bi-annually --- or ``capital indexed'' which pay interest quarterly based on a percentage of the face value adjusted for inflation.

\textbf{Stocks}\sidenote{also called shares or equity} allow their owners (or holders) to purchase part of a business. The business can periodically distribute some of its profit to its share holders in the form of {\it dividends}. Shareholders can also make money by the change in value of stocks as they are sold on a secondary market. Stocks have higher potential returns but also higher risks.

\textbf{Mutual funds} allow investors to pool their money to buy a combination of bonds and shares in a single fund. Depending on the mutual fund this has the advantage of built in diversification, as well as buying a portfolio of investments (hopefully) managed by an expert, with a lower cost in time and money than researching and purchasing each of the securities individually.

\section{Description of Advanced Securities}

\textbf{Derivatives} \index{Derivatives}\TODO{}

\textbf{Warrants} \index{Warrants}\TODO{}

\section{Portfolio Theory}
\index{Portfolio Theory}

A way to spread risk and achieve a certain balance of aggressive vs conservative is to maintain a portfolio of stock. For instance an aggressive investment portfolio may include 50\% in shares, 40\% in fixed income investments and 10\% in cash or equivalent. A conservative portfolio may have 70\% in fixed income, 20\% in equities and 10\% in cash and equivalents \cite{investing101}. Other sources such as Graham's {\it Intelligent Investor} suggest a 50-50 split between debt and equity, however note that this split changes with the circumstances of both the wider market and the risk aversion of the investor\cite{intelligentinvestor}.

Spreading the types and areas of investment is a strategy of {\it diversification}, which relies on various assets performing in different ways at different times. If any one class or category of security performs poorly, the idea is that other assets in a diverse portolio will maintain the value of the portfolio.

\section{Bond Pricing Analysis}
\index{Bond Pricing}

\TODO{} (\url{http://www.mysmp.com/bonds/bond-math.html} ?)

\section{Efficient Market Hypothesis}
\index{Efficient Market Hypothesis}

\TODO{}