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\begin{center}\textbf{Problem Set 5 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 2/26/16} \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.

\begin{enumerate}

\item Prove Theorem 3.1.8: Let $(X,\mathscr{T})$ be a topological space with $A \subseteq X$ and $U \subseteq A$.  Then $\mathrm{Cl}_A(U) = A \cap \mathrm{Cl}_X(U)$.

\item Prove Theorem 3.1.9: Let $(X,\mathscr{T})$ be a topological space with $A \subseteq X$ and $U \subseteq A$.  Then $A \cap \mathrm{Int}_X(U) \subseteq \mathrm{Int}_A (U)$.

\item Prove Theorem 3.1.11: Let $(X,\mathscr{T})$ be a topological space with $A \subseteq X$ and $U \subseteq A$.  Then $\mathrm{Bd}_A(U) \subseteq A \cap \mathrm{Bd}_X(U)$.

\item {[Baker 3.2.12]} Let $(X, \mathscr{T})$ and $(Y, \mathscr{S})$ be topological spaces and let $A \subseteq Y$.  Let $f:X \to A$ be a function.  Prove that $f$ is $\mathscr{T-S}_A$ continuous iff $f$ is $\mathscr{T-S}$ continuous as a function from $X$ to $Y$.

\item Prove Theorem 3.2.8: Let $(X,\mathscr{T})$ be a topological space and let $U \subseteq X$.  Then Int($U$) = $\{x \in X : U$ is a nghb. of $x \}$.

\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.  Don't blow off the details.

\item {[Baker 3.R.7]} Every constant function is continuous regardless of the topologies on the domain and codomain.

\item {[Baker 3.R.8]} The identity function is always continuous regardless of the topologies on the domain and codomain.

\item {[Baker 3.R.9]} If a function $f: \mathbb{R} \to \mathbb{R}$ is $\mathscr{U-U}$ continuous, then $f$ is $\mathscr{H-U}$ continuous.

\item {[Baker 3.R.10]} If a function $f: \mathbb{R} \to \mathbb{R}$ is $\mathscr{U-H}$ continuous, then $f$ is $\mathscr{U-U}$ continuous.

\item {[Baker 3.R.11]} If a function $f: \mathbb{R} \to \mathbb{R}$ is $\mathscr{H-U}$ continuous, then $f$ is $\mathscr{U-U}$ continuous.

\item {[Baker 3.R.12]} If a function $f: \mathbb{R} \to \mathbb{R}$ is $\mathscr{C-U}$ continuous, then $f$ is $\mathscr{U-U}$ continuous.

\item {[Baker 3.R.13]} Any two discrete topological spaces are homeomorphic.

\item {[Baker 3.R.14]} Any one-to-one, onto function between two discrete topological spaces is a homeomorphism.

\item {[Baker 3.R.15]} If $f$ is a homeomorphism, then $f$ is one-to-one and onto.

\item {[Baker 3.R.16]} If $f$ is a one-to-one function from one topological space onto another, then $f$ is a homeomorphism.

\item {[Baker 3.R.17]} If $(X, \mathscr{T})$ and $(Y, \mathscr{S})$ are homeomorphic topological spaces, then any one-to-one function from $X$ onto $Y$ is a homeomorphism.

\item {[Baker 3.R.18]} If $A$ and $B$ are subspaces of $\mathbb{R}^2$, where $\mathbb{R}^2$ has the usual topology, and $A$ can be deformed into $B$ by use of only elastic motions, then $A$ is homeomorphic to $B$.

\item {[Baker 3.R.19]} If $A$ and $B$ are subspaces of $\mathbb{R}^2$, where $\mathbb{R}^2$ has the usual topology, and $A$ can be deformed into $B$ by ''cutting``, then $A$ is homeomorphic to $B$.

\item {[Baker 3.R.20]} If $A$ and $B$ are subspaces of $\mathbb{R}^2$, where $\mathbb{R}^2$ has the usual topology, and $A$ can be deformed into $B$ by use of elastic motions and/or ''cutting``, then $A$ is homeomorphic to $B$.

\item {[Baker 3.R.21]} If $A$ and $B$ are subspaces of $\mathbb{R}^2$, where $\mathbb{R}^2$ has the usual topology, and $A$ can be deformed into $B$ by use of elastic motions and/or ''cutting`` and repairing the cut, then $A$ is homeomorphic to $B$.

\item {[Baker 3.R.22]} If $A$ and $B$ are subspaces of $\mathbb{R}^2$, where $\mathbb{R}^2$ has the usual topology, and $A$ can be deformed into $B$ by use of elastic motions and/or ''cutting`` and then ''repairing`` the cut so that points close to one another before the cut are also close to one another after the cut is ''repaired``, then $A$ is homeomorphic to $B$.
\end{enumerate}

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