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\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 12}}
\rhead{\large \textbf{DUE 12/10/2023}}
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{\Large\underline{\textbf{Exercises:}}}

\S $23, 24, 26$

\begin{enumerate}

\item Let $X$ be a topological space and let $\{Y_j\mid j\in J\}$ be an indexed family of connected subspaces of $X$. Suppose there exists a connected subspace $Y\subset X$ satisfying $Y\cap Y_j\neq\emptyset$ for all $j\in J$. Show that $Y\cup \bigcup_{j\in J} Y_j$ is connected.

\item Recall that $S^1=\{(x,y)\in \mathbb{R}^2\mid x^2+y^2=1\}$.
	\begin{enumerate}
	\item Show that $S^1$ is connected.
	
	\item Show that $a(x,y):=(-x,-y)$ defines a homeomorphism $a\colon S^1\to S^1$.
	
	\item Show that if $f\colon S^1\to\R$ is continuous, then there exists $(x,y)\in S^1$ satisfying $f(x,y)= f(-x,-y)$.
	\end{enumerate}

\item Let $U\subset \mathbb{R}^n$ be open and connected. Show that $U$ is path connected.

	[\textbf{Hint:} for $\mathbf{x}_0\in U$ show that the set of points $\mathbf{x}\in U$ that are connected to $\mathbf{x}_0$ by a path in $U$ is clopen.]

\item Equip $\mathbb{R}$ with the finite complement topology. Show that every subset is compact.


\item Let $X$ be a Hausdorff space. If $A,B\subset X$ are compact with $A\cap B=\emptyset$, show that there are open sets $U\supset A$ and $V\supset B$ with $U\cap V=\emptyset$.

\item Let $p\colon X\to Y$ be a closed continuous surjective map. 
	\begin{enumerate}
	\item For $U\subset X$ open, show that $p^{-1}(\{y\})\subset U$ for $y\in Y$ implies there is a neighborhood $V$ of $y$ with $p^{-1}(V)\subset U$.
	
	\item Show that if $Y$ is compact and $p^{-1}(\{y\})$ is compact for each $y\in Y$, then $X$ is compact.
	\end{enumerate}

	
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