Basis of a topology - Knowlpedia

A basis for a topology on a set $X$ is a collection $\mathcal B$ of subsets of $X$ satisfying:

Covering: for every $x\in X$ , there is $B\in\mathcal B$ with $x\in B$ .
Intersection refinement: if $x\in B_1\cap B_2$ with $B_1,B_2\in\mathcal B$ , there is $B_3\in\mathcal B$ such that $x\in B_3\subseteq B_1\cap B_2$ .

Generated topology

The topology generated by $\mathcal B$ consists of all unions of members of $\mathcal B$ , including the empty union. Its open sets are exactly the sets $U$ such that each $x\in U$ lies in some $B\in\mathcal B$ with $B\subseteq U$ .

For an already specified topology $\mathcal T$ , saying that $\mathcal B$ is a basis of $\mathcal T$ means that $\mathcal B\subseteq\mathcal T$ and every member of $\mathcal T$ is a union of members of $\mathcal B$ . This is equivalent to $\mathcal B$ generating $\mathcal T$ .

Examples

In $\mathbb{R}$ with the usual topology, the open intervals $(a,b)$ form a basis.
In a metric space, the family of open balls forms a basis for the metric-induced topology.
In a product $X\times Y$ with the product topology, the sets $U\times V$ with $U$ open in $X$ and $V$ open in $Y$ form a basis.