Products and Metrization
due by Monday, Nov 15, 2021
This assignment is based on material in lectures 20 and 21.

Let
$X$
and$Y$
be topological spaces. Prove or disprove the following. If
$X$
and$Y$
have discrete topologies then$X\times Y$
has the discrete topology.  If
$X$
and$Y$
have indiscrete topologies then$X\times Y$
has the indiscrete topology.  If
$X$
and$Y$
are connected then$X\times Y$
is connected.
 If

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.$X$
is second countable if and only if$A$
is countable.$X$
is first countable if and only if$A$
is countable.

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. 
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. 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$\varphi_\infty(a)$
.  Deduce that if $A$ is countable and $X$ is compact and firstcountable, then every sequence of functions
$\{\varphi_n\}_{n\in\mathbb{N}}$
,$\varphi_n: A \to X$
has a subsequence that converges pointwise.
 A sequence of functions

Prove or disprove that the following topologies on
$\mathbb{N}$
are metrizable. The discrete topology.
 The indiscrete topology.
 The cofinite topology.