### Subsets and Metrics

##### due by Monday, Aug 30, 2021

This assignment is based on material in lectures 5 and 6.

1. Let $X$ be a metric space, $x, y\in X$ points and $A\subset X$ a subset. Prove or disprove the following.

1. $d(A, y) \leq d(A, x) + d(x, y)$.
2. $d(x, y) \leq d(A, x) + d(A, y)$.
2. Suppose we consider the Hausdorff distance on all bounded sets (i.e., not necessarily closed). Prove or disprove the following.

1. $d_H(A, B) = 0$ if and only if $A= B$.
2. $d_H(A, B) = d_H(B, A)$.
3. $d_H(A, C)\leq d_H(A, B) + d_H(B, C)$.
3. Let $A$ be the subset of the Cantor set $X$ 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 $X$.
4. 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.