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%version 1.1,\ \   11.02.2013, Pasha Tomas oshibku nashel
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%version 1.3,\ \   25.02.2013, Kirill Elson i Dipa Pirozhkov nashli
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%version 1.4.1,\ \   01.04.2013, Topollogy!


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\begin{document}

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\listok{2}{Geometry 2: Remedial topology}

{\scriptsize
{\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra problems.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.
It's up to you to ignore handouts entirely,
because passing tests in class and having
good scores at final exams could compensate
(at least, partially) for the points obtained 
by grading handouts.

Solutions for the problems are to be explained to the 
examiners orally in the class and marked in the score sheet. 
It's better to have a written version of your solution with 
you. It's OK to share your solutions with other students, and use
books, Google search and Wikipedia, we encourage it.
The first score sheet will be distributed
February 11-th. 

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems
(except at most 2) brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.

Please keep your score sheets until the final
evaluation is given.
}

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

\subsection{Topological spaces}

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

\definition
A set of all subsets of $M$ is denoted $2^M$.
{\bf Topology} on $M$ is a collection of subsets
$S\subset 2^M$ called {\bf open subsets}, and satisfying
the following conditions.
\begin{enumerate}
\renewcommand{\labelenumi}{\arabic{enumi}.}
\item Empty set and $M$ are open

\item A union of any number of open sets is open

\item An intersection of a finite number of open subsets
is closed.
\end{enumerate}
A complement of an open set is called {\bf closed}.
A set with topology on it is called {\bf a topological space}.
{\bf An open neighbourhood} of a point is an open set containing
this point.
\ed

\definition
A map $\phi:\; M \arrow M'$
of topological spaces is called {\bf continuous}
if a preimage of each open set $U\subset M'$
is open in $M$. A bijective continuous
map is called {\bf a homeomorphism} if
its inverse is also continuos.
\ed

\exercise
Let $M$ be a set, and $S$ a set of all subsets of $M$.
Prove that $S$ defines topology on $M$. This topology
is called {\bf discrete}. Describe the set of all
continuous maps from $M$ to a given topological space.
\ez

\exercise
Let $M$ be a set, and $S\subset 2^M$ a set of two subsets:
empty set and $M$.  Prove that $S$ defines topology on $M$. This topology
is called {\bf codiscrete}. Describe the set of all continuous 
maps from $M$ to a space with discrete topology.
\ez

\definition
Let $M$ be a topological space, and $Z\subset M$
its subset. {\bf Open subsets} of $Z$ are subsets
obtained as $Z\cap U$, where $U$ is open in $M$.
This topology is called {\bf induced topology}.
\ed

\definition
{\bf A metric space} is a set $M$ equipped with 
a {\bf distance function} $d:\; M\times M \arrow \R^{\geq 0}$
satisfying the following axioms.
\begin{enumerate}
\renewcommand{\labelenumi}{\arabic{enumi}.}
\item $d(x,y)=0$ iff $x=y$.

\item $d(x,y)=d(y,x)$.

\item (triangle inequality) $d(x,y)+d(y,z)\geq d(x,z)$.
\end{enumerate}
An {\bf open ball} of radius $r$ with center in $x$
is $\{y\in M \ \ |\ \ d(x,y) < r\}$.
\ed


\definition
Let $M$ be a metric space. A subset
$U\subset M$ is called {\bf open}
if it is obtained as a union of open balls.
This topology is called {\bf induced by the metric}.
\ed

\definition
A topological space is called {\bf metrizable}
if its topology can be induced by a metric.
\ed

\exercise
Show that discrete topology can be induced by a metric,
and codiscrete cannot.
\ez

\exercise
Prove that an intersection of any collection of closed
subsets of a topological space is closed.
\ez

\definition
An intersection of all closed supersets of $Z\subset M$
is called {\bf closure} of $Z$.
\ed

\definition
{\bf A limit point} of a set $Z\subset M$ is a point
$x\in M$ such that any neighbourhood of $M$ contains
a point of $Z$ other than $x$. {\bf A limit} of a sequence
$\{x_i\}$ of points in $M$ is a point $x\in M$ such that
any neighbourhood of $x\in M$ contains all $x_i$
for all $i$ except a finite number. A sequence
which has a limit is called {\bf convergent}.
\ed

\exercise
Show that a closure of a set $Z\subset M$ is a union
of $Z$ and all its limit points.
\ez

\exercise
Let $f:\; M \arrow M'$ be a continuous map of topological spaces.
Prove that $f(\lim_i x_i) = \lim_i f(x_i)$
for any convergent sequence $\{x_i \in M\}$.
\ez

\exercise
Let $f:\; M \arrow M'$ be a map of metrizable
topological spaces, such that
$f(\lim_i x_i) = \lim_i f(x_i)$ 
for any convergent sequence $\{x_i \in M\}$.
Prove that $f$ is continuous.
\ez

\exercise[*]
Find a counterexample to the previous problem
for non-metrizable, Hausdorff topological spaces.
\ez

\exercise[**]
Let $f:\; M \arrow M'$ be a map of
countable topological spaces, such that
$f(\lim_i x_i) = f (\lim_i x_i)$ 
for any convergent sequence $\{x_i \in M\}$.
Prove that $f$ is continuous, or find a counterexample.
\ez

\exercise[*]
Let $f:\; M \arrow N$ be a bijective 
map inducing homeomorphisms
on all countable subsets of $M$.
Show that it is a homeomorphism, 
or find a counterexample.
\ez


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

\subsection{Hausdorff spaces}

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

\definition
Let $M$ be a topological space. 
It is called {\bf Hausdorff}, or {\bf separable}, if any two distinct points
$x\neq y\in M$ can be {\bf separated} by open subsets, that is,
there exist open neighbourhoods $U\ni x$
and $V\in y$ such that $U\cap V=\emptyset$.
\ed

\remark
In topology, the Hausdorff axiom is usually 
assumed by default. In subsequent handouts, 
it will be always assumed (unless stated otherwise).
\er


\exercise
Prove that any subspace of a Hausdorff space 
with induced topology is Hausdorff.
\ez

\exercise
Let $M$ be a Hausdorff topological space.
Prove that all points in $M$ are closed subsets.
\ez

\exercise
Let $M$ be a topological space, with all 
points of $M$ closed. Prove that $M$ is Hausdorff,
or find a counterexample.
\ez

\NewVedomost

\exercise
Count the number of non-isomorphic
topologies on a finite set of 4 elements.
How many of these topologies are Hausdorff?
\ez

\exercise[!]
Let $Z_1, Z_2$ be non-intersecting
closed subsets of a metrizable space $M$.
Find open subsets $U\supset Z_1$, $V\supset Z_2$
which do not intersect.
\ez


\definition
Let $M,N$ be topological spaces. {\bf Product topology}
is a topology on $M\times N$, with open sets obtained
as a union of $U\times V$, where $U$ is open in $M$
and $V$ is open in $N$.
\ed

\exercise
Prove that a topology on $X$ is Hausdorff if and only
if the diagonal $\{(x,y)\in X\times X \ \ |\ \  x=y\}$
is closed in the product topology.
\ez


\definition
Let $\sim$ be an equivalence relation on a topological
space $M$. {\bf Factor-topology} (or {\bf quotient topology})
is a topology on the set $M/\sim$ of equivalence classes
such that a subset $U\subset M/\sim$ is open
whenever its preimage in $M$ is open.
\ed

\exercise
Let $G$ be a finite group acting on 
a Hausdorff topological space $M$.\footnote{Speaking of
a group acting on a topological space, one always means
continuous action.} Prove that 
the quotient map is closed.\footnote{a {\bf closed map}
is a map which puts closed subsets to closed subsets.}
\ez

\exercise[*]
Let $\sim$ be an equivalence relation on 
a topological space $M$, and $\Gamma\subset M \times M$ {\bf its graph},
that is, the set $\{(x,y)\in M\times M \ \ |\ \ x\sim y\}$.
Suppose that the map $M \arrow M/\sim$ is open, and
the $\Gamma$ is closed in $M\times M$.
Show that $M/\sim$ is Hausdorff.
\ez

\hint
Prove that diagonal is closed
in $M\times M$.
\eh

\exercise[!]
Let $G$ be a finite group acting on 
a Hausdorff topological space $M$. Prove that $M/G$
with the quotient topology is Hausdorff.
\ez


\hint
Use the previous exercise.
\eh


\exercise[**]
Let $M=\R$, and $\sim$ an equivalence relation
with at most 2 elements in each equivalence class.
Prove that $\R/\sim$ is Hausdorff, or find a counterexample.
\ez

\exercise[*]
(``gluing of closed subsets'')
Let $M$ be a metrizable topological space,
and $Z_i\subset M$ a finite number closed subsets which do not
intersect, grouped into pairs of homeomorphic $Z_i\sim Z_i'$.
Let $\sim$ an equivalence relation
generated by these homeomorphisms.
Show that $M/\sim$ is Hausdorff.
\ez


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

\subsection{Compact spaces}

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

\definition
{\bf A cover} of a topological space $M$
is a collection of open subsets $\{U_\alpha\in 2^M\}$
such that $\bigcup U_\alpha=M$. {\bf A subcover}
of a cover $\{U_\alpha\}$ is a subset 
$\{U_\beta\}\subset \{U_\alpha\}$.
A topological space is called {\bf compact}
if any cover of this space has a finite subcover.
\ed

\exercise
Let $M$ be a compact topological space,
and $Z\subset M$ a closed subset. Show that 
$Z$ is also compact.
\ez

\exercise
Let $M$ be a countable, metrizable topological space.
Show that either $M$ contains a converging
sequence of pairwise different elements, or
$M$ contains a subset with discrete topology.
\ez


\definition
A topological space is called 
{\bf sequentially compact} if any sequence $\{z_i\}$
of points of $M$ has a converging subsequence.
\ed

\exercise
Let $M$ be metrizable a compact topological space.
Show that $M$ is sequentially compact.
\ez

\hint
Use the previous exercise.
\eh

\remark
{\bf Heine-Borel theorem} says that the 
converse is also true: any metric space which is 
sequentially compact, is also compact.
Its proof is moderately difficult 
(please check Wikipedia or any textbook
on point-set topology, metric geometry 
or analysis; ``Metric geometry'' by 
Burago-Burago-Ivanov is probably the 
best place). 

In subsequent handouts, you are allowed
to use this theorem without a proof.
\er

\exercise[*]
Construct an example of a Hausdorff topological
space which is sequentially compact,
but not compact.
\ez


\exercise[*]
Construct an example of a Hausdorff topological
space which is compact, but not sequentially compact.
\ez


\definition A {\bf topological
group} is a topological space with group operations
$G\times G \arrow G$, $x,y\mapsto xy$ and $G\arrow G$, $x\mapsto x^{-1}$
which are continuous. In a similar way, one defines
{\bf topological vector spaces}, {\bf topological
rings} and so on.
\ed

\exercise[*]
Let $G$ be a compact topological group, acting on a topological space
$M$ in such a way that the map $M\times G \arrow M$
is continuous. Prove that the quotient space is
Hausdorff.
\ez

\exercise
Let $f:\; X \arrow Y$ be a continuous map of topological
spaces, with $X$ compact. Prove that $f(X)$ is also compact.
\ez

\exercise
Let $Z\subset Y$ be a compact subset of a Hausdorff
topological space. Prove that it is closed.
\ez

\exercise
Let $f:\; X \arrow Y$ be a continuous, bijective map of topological
spaces, with $X$ compact and $Y$ Hausdorff.
Prove that it is a homeomorphism.
\ez

\definition
A topological space $M$ is called {\bf pseudocompact}
if any continuous function $f:\; M \arrow \R$
is bounded. 
\ed

\exercise
Prove that any compact topological space
is pseudocompact.
\ez

\hint
Use the previous exercise.
\eh

\exercise
Show that for any continuous function $f:\; M \arrow \R$
on a compact space there exists $x\in M$
such that $f(x)=\sup_{z\in M} f(z)$.
\ez

\exercise
Consider $\R^n$ as a metric space, with the standard
(Euclidean) metric. Let $Z\subset \R^n$ be a 
closed, bounded set (``bounded'' means ``contained
in a ball of finite radius''). Prove that
$Z$ is sequentially compact.
\ez


\exercise[**]
Find a pseudocompact Hausdorff topological space
which is not compact.
\ez

\definition
A map of topological spaces is called {\bf proper}
if a pre-image of any compact subset is always compact.
\ed

\exercise[*]
Let $f:\; X \arrow Y$
be a continuous, proper, bijective map of metrizable 
topological spaces. Prove that $f$
is a homeomorphism, or find a counterexample.
\ez

\exercise[*]
Let $f:\; X \arrow Y$
be a continuous, proper map of metrizable topological spaces.
Show that $f$ is  closed, or find a counterexample.
\ez


\end{document}