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\title{Problem Set 7: Models, compactifications, identifying spaces, characterizing subspaces}
\date{Due: Thursday, April 21}
\author{Your name:}
\begin{document}
\maketitle
Read the notes on the course website related to the subspace topology and quotient topology (sections 3 and 5, though you might enjoy 1 and 2 as well).
\begin{prob}[8]
Have you done the reading?
\end{prob}
\begin{prob}[8]
Recall that a continuous map $i: A \to X$ satisfies \emph{the universal property of the subspace topology} if for every continuous map $f: Y \to X$ such that $\im(f) \subset \im(i)$, there exists a unique continuous map $\hat{f} : Y \to A$ so that $i \circ \hat{f} = f$.
Prove that a map $f: A \to X$ is a homeomorphism on its image if and only if $f: A \to X$ satisfies the universal property of the subspace topology. (If you get stuck, go back and read the notes, which contains a proof of a very similar statement. Then try to prove this statement again. Repeat this procedure until you can understand and prove this result without consulting the notes.)
\end{prob}
\begin{prob}[9]
Let $f: X \to Y$ be a continuous surjection of topological spaces.
\begin{enumerate}
\item Give an example to show that $f$ may be an open map and at the same time not be a closed map.
\item Give an example to show that $f$ may be a closed map and at the same time not be an open map.
\item Prove that if $f$ is either open or closed, then the topology on $Y$ is equal to the quotient topology coming from the relation: $r, s \in X$ are equivalent iff $f(r) = f(s)$.
\end{enumerate}
\end{prob}
\begin{prob}[12]
This problem will define the one-point compactification and ask you to prove some theorems about it. Recall that a space $X$ is called \emph{locally compact} if for every point $p \in X$, there exists a compact neighborhood $K \subset X$ such that $p \in K$. Note that every compact space is locally compact (prove this to yourself).
\begin{enumerate}
\item Give an example of space which is locally compact and not compact.
\end{enumerate}
\noindent Recall that the \emph{one-point compactification} of a space $X$, if it exists, consists of
\begin{itemize}
\item a compact Hausdorff space $Y$
\item an embedding $i: X \hookrightarrow Y$
\end{itemize}
such that
\begin{itemize}
\item $Y \setminus i(X)$ is a set containing exactly one element (we call that element ``$\infty$" for convenience).
\item $\overline{i(X)} = Y$.
\end{itemize}
\begin{enumerate}
\addtocounter{enumi}{1}
\item Prove that if $X$ is a space with one-point compactifications $i : X \hookrightarrow Y$ and $j : X \hookrightarrow Z$ that there is a unique homeomorphism $h : Y \to Z$ such that $h \circ i = j$.
\end{enumerate}
\noindent Observe that if $X$ has a one-point compactification, then $X$ must be Hausdorff (if this is not clear, prove it to yourself!). We might also imagine that if $X$ fits inside a compact space, then it can't be that far off from being compact itself.
\begin{enumerate}
\addtocounter{enumi}{2}
\item Prove that if $X$ has a one-point compactification, then $X$ must be locally compact.
\end{enumerate}
\noindent So far in this problem you have proved that if $X$ has a one-point compactification, $X$ must be locally compact and Hausdorff. These seem to be the only ``obvious" pieces of information we can extract about $X$ if we know it has a one-point compactification. One question to ask, then, is the following:
\begin{enumerate}
\addtocounter{enumi}{3}
\item if a space $X$ is locally compact (but not compact) and Hausdorff, does it have a one-point compactification? (Hint: try using the construction we gave in class to build one.)
\end{enumerate}
\end{prob}
\begin{prob}[12]
For the following problem, the notation $(0,1)$ will refer to the open interval in the real line.
\begin{enumerate}
\item Prove that $\text{int}\D^n \cong \R^n$.
\item Prove that $(\text{int}\D^n)^{+} \cong S^n$.
\item Many of you argued that $\left( S^1 \times (0,1) \right)^{+}$ is homeomorphic to a ``pinched torus." Give a model for the pinched torus and prove that it is homeomorphic to the one-point compactification of $S^1 \times (0,1)$.
\end{enumerate}
\end{prob}
\end{document}