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\begin{center}\textbf{Problem Set 3 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 2/23/18} \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.  Four of these problems will be selected (by Jon) for grading, with each worth 5 points.

\begin{enumerate}

\item {[Baker 3.2.7]} Let $(X,\mathscr{T}), (Y,\mathscr{S})$, and $(Z,\mathscr{F})$ be topological spaces.  If the function $f:(X,\mathscr{T}) \to (Y,\mathscr{S})$ and $g:(Y,\mathscr{S}) \to (Z,\mathscr{F})$ are continuous, then $g\circ f:(X,\mathscr{T}) \to (Z,\mathscr{F})$ is continuous.\\

\item {[Baker 3.2.15]} If $A \subseteq \mathbb{R}$, then a function $f:(A,\mathscr{U}_A)\to(\mathbb{R},\mathscr{U})$ is continuous iff for each $x_0 \in A$ and each $\epsilon > 0$, there exists $\delta > 0$ such that $|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\epsilon$, for all $x \in A$.
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\item {[Baker 4.1.1]} Let $X$ and $Y$ be sets with $U$ and $A$ subsets of $X$ and $V$ and $B$ subsets of $Y$.  Prove that $(A \times B) \cap (U \times V) = (A \cap U) \times (B \cap V)$.

\item {[Baker 4.1.1.5]} Let $X$ and $Y$ be sets with $U$ and $A$ subsets of $X$ and $V$ and $B$ subsets of $Y$.  Prove that $(A \times B) \cup (U \times V) = (A \cup U) \times (B \cup V)$ or explain why not.

\item {[Baker 4.1.11]} Let $(X, \mathscr{T})$ and $(Y, \mathscr{S})$ be topological spaces with $A \subseteq X$ and $B \subseteq Y$.  Prove or disprove the following: Ext$(A \times B) = \mathrm{Ext}(A) \times \mathrm{Ext}(B)$.
  
  
\item {[Baker 4.2.8]} Let $(X_1,\mathscr{T}_1), (X_2,\mathscr{T}_2), ..., (X_n,\mathscr{T}_n)$ be topological spaces.  If $\mathscr{B}_i$ is a base for $\mathscr{T}_i$ for each $i \in \{1, 2, ..., n\}$, then the collection $\mathbb{B} = \{B_1 \times B_2 \times ... \times B_n : B_i \in \mathscr{B}_i$ for $i = 1, 2, ..., n\}$ is a base for the product space $X_1 \times X_2 \times ... \times X_n$.

\end{enumerate}


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