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\textbf{MATH 535, Topology, Homework 1}
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\underline{\textbf{1. (2+3+3 pts)}}\\
Let $G$ be a group. Our aim in this question is to first endow $\mathcal{P}(G)$ with a topology and then construct \textit{the space of subgroups of $G$} as a subspace of $\mathcal{P}(G)$.\\

For each \textbf{finite} subset $F,K \subseteq G$, let $U_{F,K}=\{S \in \mathcal{P}(G):\ F \subseteq S,\ K \subseteq S^c\}$.\\

\begin{itemize}
\item[a)] Show that the collection
\[\mathcal{B}=\{U_{F,K} \subseteq \mathcal{P}(G):\ F,K \text{ are finite subsets of } G\}\]
is a basis for a topology on $\mathcal{P}(G)$.\\
\end{itemize}

For the remainder of this question, you will consider $\mathcal{P}(G)$ with the topology $\tau_{\mathcal{B}}$ where $\tau_{\mathcal{B}}$ is the topology generated by the basis $\mathcal{B}$ in part a.\\

\begin{itemize}
\item[b)] Show that the set
\[\text{Sub}(G)=\{H \in \mathcal{P}(G):\ H \text{ is a subgroup of } G\}\]
is a closed subset of $\mathcal{P}(G)$.\\

\textbf{Hint}. Show that the complement of this set is open. In order to do that, ask yourself this: When is a subset $H \subseteq G$ not a subgroup? Is the set of $H$'s which do not contain the identity an open set? Is the set of $H$'s which do not contain the inverse of some element an open set? Is the set of $H$'s which in not closed under group multiplication an open set?\\
\end{itemize}

For the last part of this question, suppose that $G=\mathbb{Z}$ and consider $\text{Sub}(\mathbb{Z})$ with the subspace topology as a subspace of $\mathcal{P}(\mathbb{Z})$.\\

\begin{itemize}
\item[c)] Show that $\displaystyle \lim_{n \rightarrow \infty} n\mathbb{Z}=\{0\}$ in the topological space $\text{Sub}(\mathbb{Z})$.\\
\end{itemize}

\underline{\textbf{2. (2 pts)}} Let $X$ be a topological space and $A \subseteq X$. Prove that if $A$ is closed, then $A'$ is closed.\\

\underline{\textbf{3. (2 pts)}} Let $(X,\preceq)$ be a linearly ordered set. Prove that the corresponding order topology on $X$ is Hausdorff.\\

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\textbf{Solutions}
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