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).

Topological Spaces

due by Monday, Aug 16, 2021

This assignment is based on material in lectures 1 and 2.

  1. For sets $A$ and $B$, recall that $A\setminus B= \{a\in A: a \notin B\}$. Which of the following always equals $A\setminus (A \setminus B)$? Prove your answer.

    1. $A$
    2. $B$
    3. $A \cap B$
    4. $A\cup B$

  2. Let $X=\{1, 2\}$. What is the number of collections of subsets $\Omega\subset X$ that form a topology on $X$? Prove your answer.

  3. Prove that the cofinite topology on a set $X$ equals the discrete topology if and only if $X$ is finite.