2026-09-04

Topological Base

A base of a topology $(X, \tau)$ is a family of sets $\mathcal{B}$ which can enumerate any element in $\tau$ through union.

$\displaystyle x = \bigcup_{b \in \mathcal{B}}b, \forall x \in \tau$

and $\forall x \in B_1 \cap B_2, \exists B_3 \in \mathcal{B} \text{ s.t } x \in B_3 \text{ and } B_3 \subseteq B_1 \cap B_2$

~For any intersection of two basis, you can find a smaller one that is also a basis