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\begin{center}\begin{large}\textbf{Math 131 - Problem Set 6\\
Due Tuesday, Oct 16}\end{large}\end{center}



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\noindent From Munkres: 28.7abc, 29.1, 29.3, 29.6, 29.8


\begin{enumerate} 

\item Let $X$ be a compact, connected, Hausdorff topological space, and let $Y = X \cup \{\infty\}$ be its one-point compactification. Show that $Y$ is not connected. What are the connected components of $Y$ ?

\item A subset $X \subset Y$ of a topological space $Y$ is said to be \emph{locally closed} if for any point $p \in X$, there is an open subset $U\subset Y$ such that $p\in U$ and $X \cap U$ is closed in $U$.
\begin{enumerate}
\item Are the following subsets of $Y = \R^2$ locally closed? 
\begin{enumerate}
\item $X =\{(x,0) \mid x \neq 0\}$
\item $X = \{(1/n,0) \mid n \in \mathbb{Z}_+\}$
\item $X =\{(x,y) \mid x \neq 0\} \cup \{(0,0)\}$
\end{enumerate}
\item  Show that a locally closed subset $X \subset Y$ is the intersection of a closed subset of $Y$ and an open subset of $Y$.

\end{enumerate}
\end{enumerate}

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