1. Topological Spaces (20/8/2025).
  2. Bases (27/8/2025).
  3. Metrics (3/9/2025).
  4. Interval Topology (17/9/2025).
  5. Disjoint Unions (12/11/2025).

Bases

due by Wednesday, Aug 27, 2025

In each of the following, prove or disprove that $\mathcal{B}$ is a base for a topology on $X$.

  1. $X=\R$, $\mathcal{B} = \{(a, a+ \frac{1}{2^n}) : a \in \R, n \in \N\}$.
  2. $X=\R$, $\mathcal{B} = \{(a, a+ 2^n) : a \in \R, n \in \N\}$.
  3. $X=\Z$, $\mathcal{B} = \{A \subset \Z : \textrm{$A$ is finite}\}$.
  4. $X=\Z$, $\mathcal{B} = \{A \subset \Z : \textrm{$A$ is infinite}\}$.