basis (topology)

Let $(X,𝒯)$ be a topological space. A subset $ℬ$ of$𝒯$ is a basis for $𝒯$ if every member of $𝒯$ is a union of members of $ℬ$.

Equivalently, $ℬ$ is a basis if and only if whenever $U$ is open and $x\in U$ then there is an open set $V\in ℬ$ such that $x\in V\subseteq U$.

The topology generated by a basis $ℬ$ consists of exactly the unions of the elements of $ℬ$.

We also have the following easy characterization: (for a proof, see the attachment)

Proposition.

A collection of subsets $\mathrm{B}$ of $X$ is a basis for some topology on $X$ if and only if each $x\mathrm{\in }X$ is in some element $B\mathrm{\in }\mathrm{B}$ and whenever $B_{\mathrm{1}}\mathrm{,}{B}_{\mathrm{2}}\mathrm{\in }\mathrm{B}$ and $x\mathrm{\in }{B}_{\mathrm{1}}\mathrm{\cap }{B}_{\mathrm{2}}$ then there is $B_{\mathrm{3}}\mathrm{\in }\mathrm{B}$ such that $x\mathrm{\in }{B}_{\mathrm{3}}\mathrm{\subseteq }{B}_{\mathrm{1}}\mathrm{\cap }{B}_{\mathrm{2}}$.

0.0.1 Examples

1. A basis for the usual topology of the real line is given by the set of open intervals since every open set can be expressed as a union of open intervals. One may choose a smaller set as a basis. For instance, the set of all open intervals with rational endpoints and the set of all intervals whose length is a power of $1/2$ are also bases. However, the set of all open intervals of length $1$ is not a basis although it is a subbasis (since any interval of length less than $1$ can be expressed as an intersection of two intervals of length $1$).

2. More generally, the set of open balls forms a basis for the topology on a metric space.

3. The set of all subsets with one element forms a basis for the discrete topology on any set.