### 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.