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\begin{center}\textbf{Problem Set 4 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 2/10/16} \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 The empty set is a closed subset of $\mathbb{R}$ regardless of the topology on $\mathbb{R}$.
\item Any open interval is an open subset of $\mathbb{R}$ regardless of the topology on $\mathbb{R}$.
\item Any closed interval is a closed subset of $\mathbb{R}$ regardless of the topology on $\mathbb{R}$.
\item 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 If $A$ is a subset of a topological space, then $A \subseteq \mathrm{Cl}(A)$.
\item If $A$ is a subset of a topological space, then $A' \subseteq A$.
\item For any closed subset $A$ of a topological space, $A' \subseteq A$.
\item If $A$ is a subset of a topological space, then Int$(A) \subseteq A$.
\item For any subset $A$ of a topological space, Bd$(A) \subseteq A$.
\item If $A$ is a subset of a topological space, then Bd$(A) \subseteq$ Cl$(A)$.
\item If $A$ is a closed subset of a topological space, then Bd$(A) \subseteq$ Cl$(A)$.
\item If $A$ is a subset of a topological space, then Int$(A) \subseteq$ Cl$(A)$.
\item The point 1 is a limit point of the set $[0,1)$ regardless of the topology on $\mathbb{R}$.
\item The point 2 is not a limit point of the set $[0,1)$ regardless of the topology on $\mathbb{R}$.
\item For any subset $A$ of a topological space, Ext$(A) = X - A$.
\item For any closed subset $A$ of a topological space, Ext$(A) = X - A$.
\item The collection $\mathscr{B} = \{\{x\}:x \in \mathbb{R} \}$ is a base for a topology on $\mathbb{R}$. 
\item The collection $\mathscr{B} = \{\{x\}:x \in \mathbb{R} \}$ is a base for the usual topology on $\mathbb{R}$. 
\item 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 There exists a topological space $(X,\mathscr{T})$ such that there is no base for $\mathscr{T}$.
\item 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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