1. Topological Spaces (18/8/2022).
  2. Bases (25/8/2022).
  3. Metric Spaces (8/9/2022).
  4. Subsets and Continuity (15/9/2022).

Topological Spaces

due by Thursday, Aug 18, 2022

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

  2. Let $X=\Z$ and let $\Omega = \{V\subset \Z : \text{the set $\N\setminus V$ is finite}\} \cup \{\phi\}$. Prove or disprove that $\Omega$ is a topology on $X$.