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Math 70100: Functions of a Real Variable I \\
Homework 4, due Wednesday, October 1.
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{\bf Name:} Insert your name here.
\begin{questions}
\question {\em (Lang Chapter 2, problem 20)}
Recall that a space is called {\em second countable} if it has a countable base for its topology. (Lang calls this notion separable.) A topological space is {\em metrizable} if it has a metric which induces the same topology on the space.
A space is {\em normal} if it is Hausdorff and for any two disjoint closed sets $A$ and $B$ there are open sets $U$
and $V$ with $A \subset U$, $B \subset V$ and $U \cap V=\emptyset$.
Prove that a normal separable space $X$ is metrizable. (Follow the hint suggested by Lang.) (This is the {\em Urysohn Metrization Theorem}.)
\begin{sol}
Insert answer here.
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\question {\em (Royden-Fitzpatrick \S 12.1 \# 6)}
Let $X$ be a set and ${\mathcal T}$ be a topology on $X$.
Let $C(X)$ denote the collection of all continuous real-valued functions on $(X,{\mathcal T})$, and let ${\mathcal W}$ denote the weak topology induced by $C(X)$. (That is ${\mathcal W}$ is the coarsest
topology on $X$ so that every $f \in C(X)$ is continuous.)
Show that if $(X, \mathcal T)$ is normal, then the two topologies are identical.
\begin{sol}
Insert answer here.
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\question{\em (Following Rudin's Real and Complex Analysis, pp. 69)} Let $X$ be a locally compact Hausdorff space. (Recall this means that every $x \in X$ has a compact neighborhood.)
A {\em compactly supported function on $X$} is a function
$f:X \to \R$ so that there is a compact set $K \subset X$ so that
$f(x)=0$ for $x \not \in K$. We write $C_c(X)$ to denote the collection of all continuous compactly supported functions on $X$.
A function $f:X \to \R$ {\em vanishes at infinity} if for all $\epsilon>0$ there is a compact set $K \subset X$ so that
$|f(x)|<\epsilon$ for $x \not \in K$. We write $C_0(X)$ to denote the collection of all continuous functions which vanish at $\infty$.
We endow these spaces with the uniform (or sup) norm. Observe that $C_c(X) \subset C_0(X)$.
\begin{parts}
\part Show that $C_c(X)$ is dense in $C_0(X)$. ({\em Hint:} You need the version of Urysohn's lemma given in class: If $X$ is locally compact and Hausdorff, and $K \subset U \subset X$ with $K$ compact
and $U$ open, then there is a continuous $f:X \to [0,1]$ so that
$f(x)=1$ for $x \in K$ and $f(x)=0$ for $x \not \in U$.)
\part Show that $C_0(X)$ is a Banach space (i.e., that it is complete).
\end{parts}
Together, this shows that $C_0(X)$ is the metric completion of $C_c(X)$.
\end{questions}
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