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\textbf{MATH 535, Topology, Homework 4}
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\underline{\textbf{1. (2 pts)}} Let $(X,d)$ be a sequentially compact metric space and $A,B \subseteq X$ be disjoint closed subsets. Show that $\inf\{d(a,b): a \in A, b \in B\} > 0$.\\

\underline{\textbf{2. (2+3+3 pts)}} Let $(X,d)$ be a metric space.\\

\begin{itemize}
\item[(a)] Let $K \subseteq X$ be a compact subset and $x \in X$. Set $d(x,K)=\inf_{k \in K} d(x,k)$. Show that $d(x,K)=d(x,y)$ for some $y \in K$.\\
\end{itemize}
It follows from (a) that we can actually replace $\inf$ by $\min$ in the definition of $d(x,K)$ and indeed define $d(x,K)=\min_{k \in K} d(x,k)$. Using a similar argument that you used for (a), one can also show that, if $K,L \subseteq X$ are compact, then $\sup_{x \in K} d(x,L)=d(k,L)$ for some $k \in K$. Therefore, we can again replace $\sup$ by $\max$ and define $d(K,L)=\max_{x \in K} d(x,L)$ .
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Let $\mathcal{K}(X)$ denote the set of non-empty compact subsets of $X$ and consider the function $\delta_H: \mathcal{K}(X) \times \mathcal{K}(X) \rightarrow [0,\infty)$ given by
\[ \delta_H(K,L)=\max\{d(K,L),d(L,K)\}\]
You can easily check (not as a part of this homework) that $\delta_H$ is a metric on $\mathcal{K}(X)$. This metric is called the \textbf{Hausdorff metric} on $\mathcal{K}(X)$ and the topology $\tau$ generated by this metric is called the \textbf{Vietoris topology} on $\mathcal{K}(X)$.\\
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For the rest of this question, we endow $\mathcal{K}(X)$ with this topology. Before working on the rest of the question, you should get an intuitive feeling of how $\delta_H$ works. In order to do that, as an example, choose $X=\mathbb{R}^2$ and play with this. For example, find some compact sets in the ball $B_{\delta_H}([0,1]^2,1)$ and try to compute $\delta_H(S^1,[0,1]^2)$. After you get a feeling of how $\delta_H$ works, proceed.\\

\begin{itemize}
\item[(b)] Prove that if $(X,d)$ is totally bounded, then so is $(\mathcal{K}(X),\delta_H)$.\\
\end{itemize}
It turns out that if $(X,d)$ is complete, then so is $(\mathcal{K}(X),\delta_H)$. From this fact together with (b), it follows that if $(X,d)$ is compact, then so is $(\mathcal{K}(X),\delta_H)$. If you wish to see the proofs of these, you can find them in textbooks; just search the phrase ``space of compact sets".\\

\begin{itemize}
\item[(c)] Prove that $\{K \in \mathcal{K}(X): |K| \leq 535\}$ is a closed subset of $\mathcal{K}(X)$.\\
\end{itemize}



\underline{\textbf{3. (2 pts)}} Let $X$ be a topological space and $D \subseteq X$ be a dense subset. Consider the equivalence relation $\sim$ on $X$ given by
\[ x \sim y \text{ if and only if } x=y \text{ or } x,y \in D\]
In other words, $\sim$ is the equivalence relation whose quotient set is 
\[ X / \sim\ \ =\{[x]: x \in X\}=\{\{x\}: x \in D^c\} \cup \{D\}\]
Endow $X / \sim$ with the quotient topology. Show that no distinct two points in $X / \sim$ can be separated by disjoint open sets.
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\textbf{Solutions}
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