### Baire Theorem and Quotients

##### due by Monday, Nov 22, 2021

This assignment is based on material in lectures 22 and 23.

1. 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.

1. $A$ is dense.
2. The complement $\mathbb{R}^2\setminus\bar{A}$ of the closure of $A$ is a dense open set.
3. $A$ is the countable union of nowhere dense closed sets.
2. 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 $X$.

3. 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 $[0, 1]$.

4. Prove or disprove that $\mathbb{R}P^1$ is homeomorphic to $S^1$.