Subsets and Metrics
due by Monday, Aug 30, 2021
This assignment is based on material in lectures 5 and 6.

Let
$X$
be a metric space,$x, y\in X$
points and$A\subset X$
a subset. Prove or disprove the following.$d(A, y) \leq d(A, x) + d(x, y)$
.$d(x, y) \leq d(A, x) + d(A, y)$
.

Suppose we consider the Hausdorff distance on all bounded sets (i.e., not necessarily closed). Prove or disprove the following.
$d_H(A, B) = 0$
if and only if$A= B$
.$d_H(A, B) = d_H(B, A)$
.$d_H(A, C)\leq d_H(A, B) + d_H(B, C)$
.

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.$A$
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.