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\title{Problem Set 3: Limits and closures}
\date{Due: Thursday, February 18}
\author{Your name:}
\begin{document}
\maketitle
\begin{prob}[8]
Let $X$ be a topological space and $A, B \subset X$.
\begin{enumerate}
\item Show that $\overline{A \cup B} = \overline{A} \cup \overline{B}$.
\item Show that $\overline{A \cap B} \subset \overline{A} \cap \overline{B}$.
\item Give an example of $X$, $A$, and $B$ such that $\overline{A \cap B} \neq \overline{A} \cap \overline{B}$.
\item Let $Y$ be a subset of $X$ such that $A \subset Y$. Denote by $\overline{A}$ the closure of $A$ in $X$, and equip $Y$ with the subspace topology. Describe the closure of $A$ in $Y$ in terms of $\overline{A}$ and $Y$.
\end{enumerate}
\end{prob}
\begin{prob}[8]
Let $X$ be a set and let $$\t = \{ U \in \Po(X) : X \setminus U \text{ is finite, or } U = \es \}.$$
\begin{enumerate}
\item Show that $\t$ is a topology on $X$. This topology is called the \emph{cofinite topology} (or \emph{finite complement} topology).
\item Let $X$ be a set equipped with the cofinite topology. Let $A \subset X$. Describe the boundary $\partial A$ of $A$.
\item Suppose $X = \N$. To which points does the sequence $(n)_{n \in \N}$ converge?
\end{enumerate}
\end{prob}
\begin{prob}[8]
Let $(X,d)$ be a metric space. Prove that the metric topology on $X$ is Hausdorff.
\end{prob}
\begin{prob}[8]
Let $X$ and $Y$ be topological spaces. A map $f: X \to Y$ is called \emph{open} if for every open $U \subset X$, the image $f(U)$ is open in $Y$.
\begin{enumerate}
\item Consider $X \times Y$ equipped with the product topology. Show that the map $p_1: X \times Y \to X, (x,y) \mapsto x$ is both continuous and open.
\item Consider $X \coprod Y$ equipped with the sum topology. Show that the map $i_1: X \to X \coprod Y, x \mapsto (x,0)$ is both continuous and open.
\end{enumerate}
\end{prob}
\begin{prob}[12]
An \emph{equivalence relation} on a set $X$ is a subset $R \subset X \times X$ such that
\begin{itemize}
\item for each $x \in X, (x,x) \in R$.
\item for every $x,y \in X$, if $(x,y) \in R$, then $(y,x) \in R$.
\item for every $x,y,z \in X$ if $(x,y), (y,z) \in R$ then $(x,z) \in R$.
\end{itemize}
We write $x \sim_{R} y$ as an abbreviation for $(x,y) \in R$ (and sometimes just write $x \sim y$). For $x \in X$, the set $$[x] = \{ y \in X : y \sim x \}$$ is called the \emph{equivalence class} of $x$. We denote by $$X / {\sim} \, = \{ [x] : x \in X \},$$ the set of equivalence classes of elements of $X$, called the \emph{quotient of $X$ by $\sim$}.
Suppose now that $X$ is a topological space with an equivalence relation $\sim$, and consider the map $$\pi : X \to X/\sim, \quad x \mapsto [x].$$
\begin{enumerate}
\item Declare a subset $U \subset X / {\sim}$ to be open if $\pi^{-1}(U) \subset X$ is open. Show that this defines a topology on $X / {\sim}$, and that the map $\pi$ is continuous. This topology is called the \emph{quotient topology}.
\item Is the map $\pi$ always an open map? Justify your claim with proof or counterexample.
\item Let $Y$ be another topological space and let $f: X \to Y$ be a continuous map such that $f(x_1) = f(x_2)$ whenever $x_1 \sim x_2$. Show that there exists a unique map $\overline{f} : X /{\sim} \to Y$ such that $f = \overline{f} \circ \pi$, and show that $\overline{f}$ is continuous. This is called the \emph{universal property of the quotient topology}.
\item Consider $\R \coprod \R$ with the sum topology, with the equivalence relation $$(x,0) \sim (y,1) \quad \text{iff} \quad x \neq 0 \text{ and } x = y.$$ The topological space $Q = \R \coprod \R / {\sim}$ is called the \emph{line with double origin}. Which points in $Q$ are the limit of the sequence $n \mapsto [(\frac{1}{n+1}, 0)]$? Is $Q$ a Hausdorff space?
\end{enumerate}
\end{prob}
\end{document}