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

Bases

due by Thursday, Aug 25, 2022

  1. For each of the following collections of subsets of $\R^2$, prove or disprove that they form the basis for a topology on $\R^2$.

    • (a) $\{ (a, b) \times (c, d) \subset \R^2 : a, b, c, d\in \R, a < b, c < d \}$.
    • (b) $\{ (a, b) \times (c, d) \subset \R^2 : a, b, c, d\in \R, a < b, c < d - 1 \}$.
    • (c) $\{ (a, b) \times (c, d) \subset \R^2 : a, b, c, d\in \Q, a < b, c < d \}$.
    • (d) $\{ (a, b) \times \R \subset \R^2 : a, b\in \R, a < b \}$.

  2. For each pair of collections of sets in the above question which form a basis for a topology on $\R^2$, prove or disprove that the topologies they generate are equal.