Basis of a topology

Instead of listing every open set, it is enough to give a basis: the others are obtained as unions (the empty set is the union of zero members). Open balls form a basis of the topology of any metric space, and the intervals \((a,b)\) form one for the usual topology of \(\mathbb{R}\); even those with rational endpoints suffice, which makes the basis countable.

Not every family of subsets is a basis for some topology. A family \(\mathcal{B}\) of subsets of \(X\) generates a topology exactly when it covers \(X\) and, if \(x\in B_1\cap B_2\) with \(B_1,B_2\in\mathcal{B}\), there is \(B_3\in\mathcal{B}\) with \(x\in B_3\subseteq B_1\cap B_2\). In that case the generated topology \(\mathcal{T}_{\mathcal{B}}\) consists of the unions of members of \(\mathcal{B}\).

A topology has many bases: adding open sets to a basis gives another basis of the same topology. A useful way to use one is this: \(U\) is open if and only if every \(x\in U\) has some \(B\in\mathcal{B}\) with \(x\in B\subseteq U\). So, to check properties of all open sets, it is enough to check them on the members of the basis.

Bases allow us to build new topologies, such as the product topology, and to compare topologies: if each member of a basis of \(\mathcal{T}\) is a union of members of a basis of \(\mathcal{T}'\), then \(\mathcal{T}\subseteq\mathcal{T}'\).