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\title{Problem Set 4: Connectedness}
\date{Due: Thursday, February 18}
\author{Your name:}
\begin{document}
\maketitle
\begin{prob}[8]
Let $X$ be a set, and $\t_0$ and $\t_1$ topologies on $X$. If $\t_0 \subset \t_1$, we say that $\t_1$ is \emph{finer} than $\t_0$ (and that $\t_0$ is \emph{coarser} than $\t_1$).
\begin{enumerate}
\item Let $Y$ be a set with topologies $\t_0$ and $\t_1$, and suppose $\id_Y : (Y, \t_1) \to (Y, \t_0)$ is continuous. What is the relationship between $\t_0$ and $\t_1$? Justify your claim.
\item Let $Y$ be a set with topologies $\t_0$ and $\t_1$ and suppose that $\t_0 \subset \t_1$. What does connectedness in one topology imply about connectedness in the other?
\item Let $Y$ be a set with topologies $\t_0$ and $\t_1$ and suppose that $\t_0 \subset \t_1$. What does one topology being Hausdorff imply about the other?
\item Let $Y$ be a set with topologies $\t_0$ and $\t_1$ and suppose that $\t_0 \subset \t_1$. What does convergence of a sequence in one topology imply about convergence in the other?
\end{enumerate}
\end{prob}
\begin{prob}[8]
Given a space $X$, we define an equivalence relation on the elements of $X$ as follows: for all $x,y \in X$, $$x \sim y \iff \text{ there is a connected subset } A \subset X \text{ with } x,y \in A.$$ The equivalence classes are called the \emph{components} of $X$.
\begin{enumerate}
\item(0) Prove \emph{to yourself} that the components of $X$ can also be described as connected subspaces $A$ of $X$ which are as large as possible, ie, connected subspaces $A \subset X$ that have the property that whenever $A \subset A'$ for $A' $ a connected subset of $X$, $A = A'.$
\item(4) Compute the connected components of $\Q$.
\item(4) Let $X$ be a Hausdorff topological space, and $f, g: \R \to X$ be continuous maps such that for every $x\in \Q$, $f(x) = g(x)$. Show that $f = g$.
\end{enumerate}
\end{prob}
\begin{prob}[9]
Prove that no pair of the following subspaces of $\R$ are homeomorphic: $$(0,1), \quad (0,1], \quad [0,1].$$
\end{prob}
\begin{prob}[8]
Let $(X_{i})_{i \in I}$ be a family of topological spaces, and $(Y_i)_{i \in I}$ be a family of subsets $Y_i \subset X_i$. Note that the set $\prod_{i \in I} Y_i$ has two possible topologies:
\begin{itemize}
\item first give each $Y_i$ the subspace topology, and then take the product topology on the product
\item give the product the subspace topology as a subset of the product topology on $\prod_{i \in I} X_i$.
\end{itemize}
Are these two topologies the same? Prove or disprove using the universal properties of the subset and the product.
\end{prob}
\begin{prob}[12 -- problem seminar] In this problem, we will investigate the notion of convergence in the product and box topologies on spaces of functions.
\begin{enumerate}
\item Let $X$ be a space and $I$ be a set. Recall that the set of maps $X^{I}$ is also the product $\prod_{i \in I} X$, and so has a natural topology (the product topology). Let $(f_n)_{n \in \N}$ be a sequence of maps in $X^{I}$, and let $f \in X^I$. Show that $f_n \to f$ in $X^{I}$ if and only if, for every $i$, $f_n(i) \to f(i)$ in $X$. For this reason, the product topology $\t_{\prod}$ is also called the \emph{topology of pointwise convergence}.
\item Show that the topology of pointwise convergence on $\R^{\R}$ does not come from a metric.
\end{enumerate}
The \emph{topology of uniform convergence} $\t_{\infty}$ on $\R^{\R}$ is defined as follows: a subset $U \subset \R^{\R}$ belongs to $\t_{\infty}$ iff for every $f \in U$ there exists $\e > 0$ such that $$\left\{ g: \R \to \R : \underset{x \in \R}{\sup}|f(x) - g(x)| < \e \right\} \subset U.$$ Convince yourself that this is a topology. Justify to yourself the name of $\t_{\infty}$ (by figuring out what it means for a sequence to converge in $\t_{\infty}$).
\begin{enumerate} \addtocounter{enumi}{2}
\item Show that $\t_{\prod} \subset \t_{\infty} \subset \t_{\square}$
\item Show that $\t_{\prod} \neq \t_{\infty}$.
\item Show that the sequence of constant functions $x \mapsto \frac{1}{n+1}$ does not converge to $0$ in the box topology. Conclude that $\t_{\infty} \neq \t_{\square}$.
\item Find a sequence of functions $f_n \in \R^{\R}$ such that $\underset{x \in \R}{\sup} |f(x)| \geq \frac{1}{n+1}$ and that converges to the constant function $0$ in the box topology.
\end{enumerate}
\end{prob}
\end{document}