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\begin{center}\textbf{Problem Set 3 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 2/5/20} \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 Ch 2 R1]} The empty set is a closed subset of $\mathbb{R}$ regardless of the topology on $\mathbb{R}$.\\

\item {[Baker Ch 2 R2]} Any open interval is an open subset of $\mathbb{R}$ regardless of the topology on $\mathbb{R}$.\\

\item {[Baker Ch 2 R3]} Any closed interval is a closed subset of $\mathbb{R}$ regardless of the topology on $\mathbb{R}$.\\

\item {[Baker Ch 2 R4]} A half-open interval of the form $[a,b)$ is neither an open set nor a closed set regardless of the topology on $\mathbb{R}$.\\

\item {[Baker Ch 2 R5]} If $A$ is a subset of a topological space, then $A \subseteq \mathrm{Cl}(A)$.\\

\item {[Baker Ch 2 R6]} If $A$ is a subset of a topological space, then $A^\prime \subseteq A$.\\

\item {[Baker Ch 2 R7]} For any closed subset $A$ of a topologoical space, $A^\prime \subseteq A$.\\

\item {[Baker Ch 2 R8]} If $A$ is a subset of a topological space, then $\mathrm{Int}(A) \subseteq A$.\\

\item {[Baker Ch 2 R9]} For any subset $A$ of a topological space, $\mathrm{Bd}(A) \subseteq A$.\\

\item {[Baker Ch 2 R10]} If $A$ is a subset of a topological space, then $\mathrm{Bd}(A) \subseteq \mathrm{Cl}(A)$.\\

\item {[Baker Ch 2 R11]} If $A$ is a closed subset of a topological space, then $\mathrm{Bd}(A) \subseteq A$.\\

\item {[Baker Ch 2 R12]} If $A$ is a subset of a topological space, then $\mathrm{Int}(A) \subseteq \mathrm{Cl}(A)$.\\

\item {[Baker Ch 2 R13]} The point 1 is a limit point of the set $[0,1)$ regardless of the topology on $\mathbb{R}$.\\

\item {[Baker Ch 2 R14]} The point 2 is not a limit point of the set $[0,1)$ regardless of the topology on $\mathbb{R}$.\\

\item {[Baker Ch 2 R15]} For any subset $A$ of a topological space, $\mathrm{Ext}(A) = X-A$.\\

\item {[Baker Ch 2 R16]} For any closed subset $A$ of a topological space, $\mathrm{Ext}(A) = X-A$.\\

\item {[Baker Ch 2 R17]} The collection $\mathscr{B} = \{\{x\}:x \in \mathbb{R}\}$ is a base for a topology on $\mathbb{R}$.\\

\item {[Baker Ch 2 R18]} The collection $\mathscr{B} = \{\{x\}:x \in \mathbb{R}\}$ is a base for the usual topology on $\mathbb{R}$.\\

\item {[Baker Ch 2 R19]} In a space $(X,\mathscr{T})$ any collection of open sets whose union equals $X$ and that is closed under finite intersection is a base for $\mathscr{T}$.\\

\item {[Baker Ch 2 R20]} There exists a topological space $(X,\mathscr{T})$ such that there is no base for $\mathscr{T}$.\\

\item {[Baker Ch 2 R21]} There exists a topological space $(X,\mathscr{T})$ for which there is more than one base for $\mathscr{T}$.\\

\end{enumerate}


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