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\begin{center}\textbf{Problem Set 4 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 2/24/20} \end{center}

\noindent You are expected to do the following problems to a high standard (i.e., at least well enough to be published in a textbook) for full credit.  Five of these problems will be selected (by Jon) for grading, with each worth 4 points.

\begin{enumerate}

\item {[Baker Th 3.1.7]} Let $(X,\mathscr{T})$ be a topological space with $A \subseteq X$ and $U \subseteq A$.  The set $U$ is $\mathscr{T}_A$-closed iff $U = W \cap A$ for some $\mathscr{T}$-closed set $W$.\\

\item {[Baker Th 3.1.14]} Let $A$ be a subset of the space $(X,\mathscr{T})$.  The set $A$ is $\mathscr{T}$-open iff $\mathscr{T}_A \subseteq \mathscr{T}$.\\

\item {[Baker Th 3.1.13]} If $\mathscr{B}$ is a base for a topological space $(X, \mathscr{T})$ and $A \subseteq X$, then the collection $\{B \cap A : B \in \mathscr{B}\}$ is a base for $(A, \mathscr{T}_A)$.\\

\item {[Baker Th 3.2.14]} Let $(X, \mathscr{T})$ and $(Y,\mathscr{S})$ be topological spaces and let $A \subseteq X$. If $f:(X,\mathscr{T}) \rightarrow (Y,\mathscr{S})$ is continuous, then $f|_A:(A,\mathscr{T}_A) \rightarrow (Y,\mathscr{S})$ is continuous.\\

\item {[Baker 3.3.9]} Let $(a,b)$ and $(c,d)$ be open intervals.  Prove that the spaces $((a,b),\mathscr{U}_{(a,b)})$ and $((c,d),\mathscr{U}_{(c,d)})$ are homeomorphic.\\

\item Partition the spaces 1 2 3 4 5 6 7 8 9 0 into mutually disjoint collections of hemeomorphic spaces such that, if two spaces belong to different collections, then they are not homeomorphic.\\

\item Partition the spaces a b c d e f g h i j k l m n o p q r s t u v w x y z into mutually disjoint collections of hemeomorphic spaces such that, if two spaces belong to different collections, then they are not homeomorphic.\\

\item {[Baker Th 4.1.10]} Let $(X,\mathscr{T})$ and $(Y,\mathscr{S})$ be topological spaces.  If $A$ and $B$ are closed subsets of $X$ and $Y$, respectively, then $A \times B$ is a closed subset of $X \times Y$.
\end{enumerate}


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