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\begin{document}
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Math 70100: Functions of a Real Variable I \\
Homework 2, due Wednesday, September 17th.
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{\bf Name:} Insert your name here.
\begin{questions}
\question {\em (Modified from Folland 4.1.13)} Suppose $X$ is a topological space and $A \subset X$ is dense.
Prove that if $U \subset X$ is open, then $\bar U=\overline{U \cap A}$, where $\bar \cdot$ denotes closure.
\begin{sol}
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\question (From \href{http://www.math.qc.edu/~zakeri/mat701/h2.pdf}{Zakeri's Homework 2}) Give a direct proof that the interval $[0,1]$ is compact. \\
({\em Hint:} Let ${\mathcal U}$ be an open cover. Define
$$S=\{x \in [0,1]~:~\text{$[0,x]$ is covered by finitely many $U \in {\mathcal U}$}\}.)$$
Prove that $S=[0,1]$.)
\begin{sol}
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\question {\em (Modified from Lang II.5.1a)} Let $X$ and $Y$ be compact Hausdorff topological spaces. Prove that $f:X \to Y$ is continuous if and only if its graph is closed in $X \times Y$. (The {\em graph} of $f$ is the set
$$\Gamma=\{(x,y) \in X \times Y ~:~ y=f(x)\}.)$$
({\em Remark:} More generally, the result is true if $X$ is just a topological space and $Y$ is a compact Hausdorff space. This is the {\em closed graph theorem}.)
\begin{sol}
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\question {\em (Modified from Lang II.5.1b)} A function $f:X \to Y$ between metric spaces is {\em uniformly continuous} if for all $\epsilon>0$, there is a $\delta>0$ so that $d_X(x_1,x_2)<\delta$
implies $d_Y\big(f(x_1),f(x_2)\big)<\epsilon$ for all $x_1,x_2 \in X$.
Let $Y$ be a complete metric space and $X$ be a metric space.
Let $A \subset X$. Let $f:A \to Y$ be uniformly continuous,
and let $\bar A \subset X$ denote the closure of $A$. Show that there exists a unique extension of $f$ to a continuous map $\bar f:\bar A \to Y$, and show that $\bar f$ is uniformly continuous. (You may assume that $X$ and $Y$ are subsets of Banach spaces if you wish, in order to write the distance function in terms of the absolute value sign.)
\begin{sol}
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\question {\em (Lang II.5.12)} Let $U$ be an open subset of a normed vector space. Show that $U$ is connected if and only if $U$ is path (or arcwise) connected. (Recall that if a topological space is path connected, then it is connected. See Proposition 2.7. You do not need to prove this.) ({\em Hint}: define the notion of a path-component, which is analogous to the notion of connected component.)
\begin{sol}
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\question The closed topologist's sine curve is
$$T=\big\{(x,\sin \frac{\pi}{x})~:~0 < x \leq 1 \big\}\cup \{(0,y)~:~y \in [-1,1]\}.$$
Show that $T$ is connected but not path connected.
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\end{questions}
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