K. Jänich, Topology. Chapter I: Fundamental Concepts. 4: Bases and Subbases

Definition 1 Let \(X\) be a topological space. A basis for the topology on \(X\) is a set \(\mathcal{B}\) of open sets such that every open set is the union of a subset of \(\mathcal{B}\).

Example 2 Let \(X\) and \(Y\) be topological spaces. Then \( \{ U \times V : U \text{ is open in } X, V \text{ is open in } Y \} \) is a basis for the product topology on \( X \times Y \).

Example 3 The open balls form a basis for the topology on a metric space.

Example 4 The open balls with rational centres and rational radii form a basis for the standard topology on \(\mathbb{R}^n\). Hence this topology has a countable basis.

Definition 5 Let \(X\) be a topological space. A subbasis for the topology on \(X\) is a set \(\mathcal{S}\) of open sets such that every open set is a union of finite intersections of elements of \(\mathcal{S}\) (where we take the intersection of zero sets to be \(X\)).

Proposition 6 Let \(X\) be a set and \( \mathcal{S} \subseteq \mathcal{P} X \). Then there exists a unique topology on \(X\) with respect to which \(\mathcal{S}\) is a subbasis.

Proof The topology is the set of all unions of finite intersections of members of \(\mathcal{S}\). It is easy to check that this satisfies the definition of a topology.

We call this topology the topology generated by \(\mathcal{S}\). It is the coarsest topology that includes \(\mathcal{S}\).

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