topological basis

Overview

Basis are usually an easier way to talk about a topology It gives us an easy way to compare topolgies

Definition

For a set X with a topology $\tau$, a set of subsets of X (let it be $\mathcal{B}$) is a basis for $\tau$ if

1. $\forall x\in X,\exists B\in \mathcal{B}$ where $x\in B$ Every element of X is covered by the basis
2. For 2 basis elements $B_1,B_2$ if $x\in B_1\cap B_2$ then there is some basis element $B_3$ such that $x\in B_3\subset B_1\cap B_2$ If x is in more than one basis, there is a third basis that is in the intersection (it is smaller than both of the original ones).