Baire Theorem and Quotients
due by Monday, Nov 22, 2021
This assignment is based on material in lectures 22 and 23.

Let
$A\subset\mathbb{R}^2$
be the subset$A = \{(x, y)\in \mathbb{R}^2: x\in \mathbb{Q}\}$
. Prove or disprove the following.$A$
is dense. The complement
$\mathbb{R}^2\setminus\bar{A}$
of the closure of $A$ is a dense open set. $A$
is the countable union of nowhere dense closed sets.

Let
$X$
be a topological space. Let$\sim$
be the equivalence relation on$X$
such that$$x\sim y\iff x = y.$$
Prove or disprove that$X/\sim$
is homeomorphic to$X$
. 
Let
$X$
be the quotient of$[0, 1]$
by the equivalence relation$\sim$
generated by$x\sim (1  x)$
for all$x\in [0, \frac{1}{2}]$
. Prove or disprove that$X$
is homeomorphic to$[0, 1]$
. 
Prove or disprove that
$\mathbb{R}P^1$
is homeomorphic to$S^1$
.