Basis of a topology

A basis of a topology on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ such that:

1. for every $x\in X$ there exists $B\in\mathcal{B}$ with $x\in B$,
2. if $x\in B_1\cap B_2$ with $B_1,B_2\in\mathcal{B}$, then there exists $B_3\in\mathcal{B}$ such that $x\in B_3\subseteq B_1\cap B_2$.

The topology generated by $\mathcal{B}$ is the collection of all unions of elements of $\mathcal{B}$; these unions are exactly the open sets . Bases are a standard way to specify a topology efficiently and to test continuity using only basic open sets.

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.