Compactness for Metrics and Compactification
due by Monday, Nov 1, 2021
This assignment is based on material in lectures 18 and 19.

Consider the cover of
$X = [0, 1]$
by sets of the form$X\cap (a, a + \frac{1}{3})$
for$a\in \mathbb{R}$
. Find a Lesbegue number for this cover (and prove that it is a Lesbesgue number). 
Let
$X=[0, 1]\times[0, 1]\subset\mathbb{R}^2$
. For which positive real numbers$d> 0$
is there a constant$C = C_d > 0$
such that for every$\varepsilon\in (0, 1)$
there is an$\varepsilon$
net of size at most$C_d(\frac{1}{\varepsilon})^d$
in$X?$
Prove your answer. 
Let $X$ be a topological space and let $X^*$ be its onepoint compactification. Prove or disprove the following.
 If $X$ is compact then $X^*$ is homeomorphic to $X$.
 If $X$ is compact then $X^*$ is not connected.
 If $X$ is connected then $X^*$ is connected.

Give an example of a topological space
$X$
that is not locally compact and prove that it is not locally compact (you may wish to consider the SNCF metric).