Let \(X\) be a topological space . We call a collection \(\mathcal{B}\) of subsets of \(X\) basis for the topology \(X\) if
- every element of \(\mathcal{B}\) is open and
- every open subset of \(X\) is the union of some collection of elements of \(\mathcal{B}\).
Remark
- For metric spaces \(B_{\varepsilon}(x)\) for arbitrary \(\varepsilon\) and \(x\) form a basis.
- It is possible to define topologies by specifying a family of subsets.