Suppose \(X\) is a topological space and \(\mathcal{B}\) is a basis for its topology. Then \(O\subseteq X\) is open if and only if for every \(p\in U\), there exists an element \(B\in \mathcal{B}\) such that \(p\in B\subseteq U\).
The “right hand side” is called basis criterion.
Proof
If \(O\) is open the claim follows by the definition of a basis.
Otherwise, we write \(O\) is the union of elements in \(\mathcal{B}\) and thus \(O\) is open.