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\begin{center}{\bf Homework 5, due Thursday, October 11, 2012}\end{center}
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Do any 5 of the 8 problems.
Each problem is worth 20 points. Solutions will be graded for correctness, clarity and style.
\begin{itemize}
\item[(1)] Let $X$ be a topological space. If $C$ is a finite subset of $X$, show that $C$ is compact.
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\item[(2)] Let $X$ be a set with the discrete topology. If $C\subseteq X$ is compact,
show that $C$ is finite.
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\item[(3)] Let $X$ be a topological sapce. Show that $X$ is a $T_1$-space if and only if each point of $X$ is a closed set.
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\item[(4)] Give a direct proof that a metric space $(X,d)$ is Hausdorff. (Do not for example use the fact that
a metric space is a $T_3$-space and every $T_3$-space is a $T_2$-space.)
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\item[(5)] Let $f:X\to Y$ be a continuous bijective map of topological spaces. Note that since $f$ is bijective
we can define the inverse function $f^{-1}:Y\to X$ as $f^{-1}(y)=x$ whenever $f(x)=y$.
If $X$ is compact and $Y$ is Hausdorff, show that $f^{-1}$ is also continuous.
[Aside: when a continuous bijective map $f:X\to Y$ has
a continuous inverse, we say that the map $f$ is a
{\it homeomorphism} and that $X$ and $Y$ are {\it homeomorphic}.]
{\it homeomorphism} and that $X$ and $Y$ are {\it homeomorphic}.]
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\item[(6)] Give an example of a continuous bijective map $f:X\to Y$ where $X$ is compact
but $Y$ is not Hausdorff and where $f^{-1}$ is not continuous.
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\item[(7)] Let $X={\bf R}$ have the standard topology, and let $Y=(0,\infty)\subset X$ have the subspace topology.
Show that $X$ and $Y$ are homeomorphic. (I.e., find a continuous bijective map $f:X\to Y$ with a continuous inverse. You do not need to prove that your $f$ is continuous, bijective
or has a continuous inverse, but it should be obvious to a Calc I student that
the $f$ that you pick is continuous, bijective
and has a continuous inverse.)
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\item[(8)] Let $(X,d)$ be a metric space and let $a\in X$.
Define $f:X\to{\bf R}$ by $f(x)=d(a,x)$ for all $x\in X$.
If ${\bf R}$ has the standard topology, show that $f$ is continuous.
\end{itemize}
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