### Products and Metrization

##### due by Monday, Nov 15, 2021

This assignment is based on material in lectures 20 and 21.

1. Let $X$ and $Y$ be topological spaces. Prove or disprove the following.

1. If $X$ and $Y$ have discrete topologies then $X\times Y$ has the discrete topology.
2. If $X$ and $Y$ have indiscrete topologies then $X\times Y$ has the indiscrete topology.
3. If $X$ and $Y$ are connected then $X\times Y$ is connected.
2. Let $A$ be a set and let $X=\prod_{a\in A}\mathbb{R}$ be the product of copies of $\mathbb{R}$ indexed by $A$. Prove or disprove the following.

1. $X$ is second countable if and only if $A$ is countable.
2. $X$ is first countable if and only if $A$ is countable.
3. Let $D$ be the discrete topology on a set with $2$ points and let $X= \prod_{n\in\mathbb{N}} D$ be the product of countably many copies of $D$. Prove or disprove that $X$ is homeomorphic to the Cantor set.

4. Let $A$ be a set and $X$ a topological spaces. Observe that elements $(\varphi_a)_{a \in A} \in \prod_{a \in A} X$ can be identified with functions $\varphi: A \to X$, $\varphi(a) = \varphi_a$. Using this identification, prove the following.

1. A sequence of functions $\{\varphi_n\}_{n\in\mathbb{N}}$, $\varphi_n: A \to X$ converges to a function $\varphi_\infty$ in the product topology (when the functions are regarded as elements of $\prod_{a \in A} X$) if and only if the functions converge to $\varphi_\infty$ pointwise, i.e., for all $a\in A$, $\varphi_n(a)$ converges to $\varphi_\infty(a)$.
2. Deduce that if $A$ is countable and $X$ is compact and first-countable, then every sequence of functions $\{\varphi_n\}_{n\in\mathbb{N}}$, $\varphi_n: A \to X$ has a subsequence that converges pointwise.
5. Prove or disprove that the following topologies on $\mathbb{N}$ are metrizable.

1. The discrete topology.
2. The indiscrete topology.
3. The cofinite topology.