Subsets and Continuity
due by Thursday, Sep 15, 2022

Let
$A$
be the subset of the Cantor set$X$
(recall this is defined as$\{0, 1\}^\mathbb{N}$
with an ultrametric) with$A$
consisting of sequences$\{a_n\}$
such that$a_n= 0$
for all but finitely many$n$
. Prove or disprove the following.$A$
is open in$X$
.$A$
is closed in$X$
.$A$
intersects every open set in$X$
.

Prove or disprove the following.
 A subspace of a space with discrete topology has discrete topology.
 A subspace of a space with indiscrete topology has indiscrete topology.
 A subspace of a space with cofinite topology has cofinite topology.

Let
$Y$
be a topological space such that for all topological spaces$X$
, every map$f: X\to Y$
is continuous. Prove that$Y$
has the indiscrete topology.