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{{\noindent Math 5510} \hfill {Homework 5--  Due  Wednesday 11/1/17}
\hfill {\ \ \ \ }\hfill}



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 \hrule



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\begin{enumerate}

\item (2pts) Let $X$ be a compact Hausdorff space, let $A,B\subset X$ be disjoint closed sets.  Prove that there exist disjoint open sets $U,V\subset X$ with $A\subset U$ and $B\subset V$.
\vskip.3cm

\item (4pts)  Let $(X,d)$ be a metric space and let $A\subset X$, let $x\in X$.  Define the \emph{distance between $x$ and $A$}, denoted $d(x,A)$, by
$$
d(x,A)= \inf\{ d(x,y) \ | \ y\in A\}
$$

\begin{enumerate}
\item Prove that $f(x) = d(x,A)$ is a continuous function of $x$.
\vskip.15cm 
\emph{Suggestion} Prove that $f$ is Lipschitz.
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\item 
Prove that $d(x,A) = 0$ if and only if $x\in\overline{A}$.
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\item
Observe that it follows from (a) and (b) that if $A\subset X$ is closed, there exists a continuous function $f:X\to [0,\infty)$ such that $f(x) = 0$ if and only if $x\in A$.  
\vskip.15cm
 Prove that if $A,B\subset X$ are disjoint closed subsets, there exists a function $g:X\to [0,1]$ such that $g(x) = 0 \Longleftrightarrow x\in A$ and $g(x) = 1\Longleftrightarrow x\in B$.  
 \vskip.15cm
 Conclude, from knowing that $g$ exists, that if $A,B\subset X$ are disjoint closed sets, there exist disjoint open sets $U,V\subset X$ with $A\subset U$ an $B\subset V$.
 \vskip.15cm
\emph{Suggestion} To find $g$, use some fraction with numerator and denominator simple expressions in $d(x,A)$ and $d(x,B)$.
\end{enumerate}
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\item (2 pts) Recall that a topological space is said to be \emph{locally connected} if and only if it has a basis consisting of connected open sets. 
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  \begin{enumerate}
  \item Prove that if $X$ is locally connected, then its connected components are open.
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  \item Prove that if $X$ is locally connected and $Y$ is the space of its connected components, with the quotient ( $=$ identification) topology, then $Y$ is discrete.


  
\end{enumerate}

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\item  (2 pts)  A topological space $X$ is said to be \emph{totally disconnected} if and only if, for all $x\in X$, the connected component of $x$ is simply $\{x\}$.
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\begin{enumerate}
\item Prove that, if for all $x,y\in X$ there exists a continuous function $\phi:X\to \{0,1\}$ (where $\{0,1\}$ has the discrete topology) such that $\phi(x)\ne \phi(y)$, then $X$ is totally disconnected.
\vskip.2cm
\item Prove that $\{0,1\}^\N$, with the product topology, is totally disconnected.  (Recall that product topology makes sense since $\{0,1\}^\N = \prod_1^\infty \{0,1\}$ )
  \end{enumerate}  
  
    \end{enumerate}  
   
 

                

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