### Subsets and Continuity

##### due by Thursday, Sep 15, 2022
1. Let $A$ be the subset of the Cantor set $X$ (recall this is defined as $\{0, 1\}^\mathbb{N}$ with an ultra-metric) with $A$ consisting of sequences $\{a_n\}$ such that $a_n= 0$ for all but finitely many $n$. Prove or disprove the following.

1. $A$ is open in $X$.
2. $A$ is closed in $X$.
3. $A$ intersects every open set in $X$.
2. Prove or disprove the following.

1. A subspace of a space with discrete topology has discrete topology.
2. A subspace of a space with indiscrete topology has indiscrete topology.
3. A subspace of a space with cofinite topology has cofinite topology.
3. 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.