1. Topological Spaces (16/8/2021).
  2. Bases and Metrics (23/8/2021).
  3. Subsets and Metrics (30/8/2021).
  4. Continuity (13/9/2021).
  5. More Topologies and Homeomorphisms (20/9/2021).
  6. Connectedness and Applications (11/10/2021).
  7. Separation Properties (18/10/2021).
  8. Countability and Compactness (25/10/2021).
  9. Compactness for Metrics and Compactification (1/11/2021).
  10. Products and Metrization (15/11/2021).
  11. Baire Theorem and Quotients (22/11/2021).

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 $S^1$.