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\listok{9}{GEOMETRY 9: Connectedness}

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\begin{opredelenie}
Let $M$ be a topological space.
A closed and open at the same time 
subset $W\subset M$ is called {\bf clopen}. 
$M$ without proper clopen subsets is called
{\bf connected}.
A subset $Z\subset M$ is called {\bf connected} if
it is connected in the induced topology. 
\end{opredelenie}

\begin{zadacha}
Is $\R$ connected?
\end{zadacha}

\begin{zadacha}[!]
Let $X$, $Y$ be connected. Show that $X\times Y$ is connected.
\end{zadacha}

\begin{ukazanie}
Let 
$U\subseteq X\times Y$ be clopen.
Consider $U\cap X\times \{y\}$. Show that
$X\times \{y\}$ (in induced topology) is homeomorphic to 
$X$, and $U\cap X\times \{y\}$ is clopen there. 
\end{ukazanie}

\begin{zadacha}
Is $\R^n$ connected (in its natural topology)?
\end{zadacha}

\begin{zadacha}
Assume that it is possible to connect any two points $x, y$ in $M$
by a path, that is, to find a continuous mapping 
$[0,1] \stackrel \phi \arrow M$ satisfying
$\phi(0)=x$, $\phi(1)=y$. Show that
$M$ is connected.
\end{zadacha}

\begin{zamechanie} Such an $M$ is called {\bf path-connected}.
 \end{zamechanie}

\begin{zadacha}
Remove a point from a circle or the plane.
Show that the result is connected.
\end{zadacha}

\begin{zadacha}[!]
\begin{enumerate} \item Remove a finite number of points from $\R^2$.
Show that the result is connected.

\item Remove a point from an interval.
Show that the result is not connected.
\end{enumerate}
\end{zadacha}

\begin{zadacha}[!]
Show that the following spaces are not homeomorphic to each other:
$\R$, $\R^2$, the circle.
\end{zadacha}

\begin{zadacha}[!]
Show that the following spaces are not homeomorphic to each other:
closed interval, half-open interval, open interval.
\end{zadacha}

\begin{zadacha}
Let
$f:\; X \arrow Y$ be continuous and  $X$ be connected.
Show that $f(X)$ is connected.
\end{zadacha}

\begin{zadacha}[!]
Let $U\subseteq [0,1]$ be connected.
Show that $U$ is either a
closed interval, or a half-open interval, or an open interval.
\end{zadacha}

\begin{zadacha}
Let
$f:\; X \arrow \R$ be continuous and  $X$ be connected.
Assume that $f$ takes positive as well as negative values.
Show that $f(x)=0$ for some $x\in X$.
\end{zadacha}

\begin{zadacha}[*]
Let $M$ be a connected metrizable countable topological space.
Show that $M$ consists of one point.
\end{zadacha}

\begin{zadacha} 
Show that the union of two connected subsets of a topological space
$M$ is connected, provided that their intersection is nonempty.
\end{zadacha}


\begin{zadacha}[!]
Let $x\in M$ and  $W$ be the union of all the connected subsets of $M$
containing $x$. Show that $W$ is connected.
\end{zadacha}


\begin{opredelenie}
In such a situation $W$ is called {\bf the connected component}
of $x$ (or just {\bf a connected component}).
\end{opredelenie}

\begin{zadacha}
Show that $W\subset M$ is a connected component if and only if
any connected subset containing $W$ coincides with $W$.
\end{zadacha}

\begin{zadacha}
Show that $M$ is the disjoint union of its connected components.
\end{zadacha}

\begin{zadacha}
Show that each connected component of $M$ is closed.
\end{zadacha}

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\subs{Totally disconnected spaces}
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\begin{opredelenie}
A topological space
$M$ is called {\bf totally disconnected}
if each connected component of $M$ consists of one point.
\end{opredelenie}

\begin{zadacha}
Show that $\Q$, the space of rational numbers, 
in the topology induced by $\R$, is totally disconnected,
but not discrete.
\end{zadacha}

\begin{zadacha}[*]
Show that $\Q_p$, the space of $p$-adic numbers, is totally disconnected.
\end{zadacha}

\begin{zadacha}[*]
Show that the product of totally disconnected spaces is totally disconnected.
\end{zadacha}

\begin{zadacha} 
Let $S$ be a subbase in a Hausdorff topological space $M$, and all the
elements of $S$ clopen. Show that $M$ is totally disconnected. 
\end{zadacha}

\begin{zadacha}[!]
Consider the set $\{0, 1\}$ equipped with the discrete topology.
Let $\{0, 1\}^I$ be the product of 
$I$ copies of  $\{0, 1\}$ with Tychonoff topology, 
with $I$ being an arbitrary index set.
Show that $\{0, 1\}^I$ is totally disconnected.
\end{zadacha}

\begin{ukazanie}
Use the preceding problem.
\end{ukazanie}

\begin{zadacha}[*]
Let $M$ be Hausdorff topological space, 
$M_1$ be the sets of connected components of $M$, and
$M\overset{\pi}{\arrow} M_1$ the natural projection (each point
is mapped to its connected component).
On $M_1$ introduce the following topology:
$U\subset M_1$ open if $\pi^{-1}(U) \subset M$ is open.
Show that $M_1$ is totally disconnected.
Show that any continuous mapping
$M\overset{\pi_2}{\arrow} M_2$
from $M$ to a totally disconnected space $M_2$
can be written as a composition of continuous mappings
$M\overset{\pi}{\arrow} M_1 \arrow  M_2$.
\end{zadacha}

\begin{ukazanie}
If $S\subset M_1$ is connected then the preimage
$\pi^{-1}(S)$ is connected, too. Indeed, if
$W\subset \pi^{-1}(S)$ is clopen then 
$W= \pi^{-1}(W_1)$ (if $W$ intersects a connected component of 
$M$, then $W$ contains it). Thus $W_1$ is clopen.
\end{ukazanie}

\begin{zadacha} 
Let $U$ be an open subset of a compact Hausdorff space
and a collection of closed subsets
$\{K_i\}$, so that their intersection is conatined in $U$.
Show that $\{K_i\}$ contains a finite subcollection
so that their intersection is contained in $U$.
\end{zadacha}

\begin{zadacha}[*] 
Let $M$ be a totally disconnected compact Hausdorff space.
Show that, for each point $x\in M$, the intersection of all the
clopen subsets of $M$ containing $x$ is $\{x\}$.
\end{zadacha}

\begin{ukazanie}
Let $P$ be the intersection of the clopen subsets containing $x$.
Obviously $P$ is closed. Show that $P$ is either $\{x\}$ or disconnected.
In the latter case $P$ is the disjoint union of two nonempty 
closed subsets $P_1$, $P_2$. As T4 holds in $M$ (show this),
find for 
$P_1$, $P_2$ nonintersecting open neighbourhoods $U_1$, $U_2$.
Derive from the preceding problem that $U_1 \cup U_2$ contains
a clopen subset $W\subset M$ containing $x$.
Show that $W\cap U_i$ are clopen, and derive from this 
that $P=\{x\}$.
\end{ukazanie}

\begin{zadacha}[*]
Let $M$ be a totally disconnected compact Hausdorff space.
Show that the clopen subsets form a base of the topology of $M$.
\end{zadacha}


\begin{ukazanie}
Let $U\subset M$ be open and 
$x\in U$. 
For each point in  $M\backslash U$ pick a clopen neighbourhood that
does not contain $x$ (show that this is always possible).
This is a cover $\{U_\alpha\}$ of
$M\backslash U$. As $M\backslash U$ is compact,
$\{U_\alpha\}$ contains a finite subcover $U_1, ... U_n$. 
Show that the complement to 
$\cup U_i$ is clopen, contains $x$, and is contained in $U$.
\end{ukazanie}

\begin{zadacha}[*]
Let $M$ be a totally disconnected compact Hausdorff space.
Let $x, y \in M$ be two distinct points.
Show that $M$ admits a continuous mapping to 
$\{0, 1\}$ (with discrete topology) such that $x$ goes to $0$ and 
$y$ goes to $1$.
\end{zadacha}

\begin{zadacha}[*]
Let $M$ be a totally disconnected compact Hausdorff space.
Let $I$ be the set of all continuous mappings from 
$M$ to $\{0, 1\}$. Define a natural mapping
$M \arrow \{0, 1\}^I$. Show that it is a continuous embedding, 
and that the image of $M$ is closed.
\end{zadacha}

\begin{zadacha}[*]
Let $M$ be a compact Hausdorff space. Show that the following
statements are equivalent.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $M$ is totally disconnected.
\item $M$ can be embedded into $\{0, 1\}^I$
for some set $I$ of indices.
\end{enumerate}
\end{zadacha}

\begin{zamechanie}
Recall that if a compact
$M$ admits a continuous injective mapping
$f:M \to X$ into a Hausdorff space $X$ then $f$ is a homeomorphism
between $M$ and $f(M) \subset X$ with induced topology.
\end{zamechanie}

\end{document}