basis (topology)

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

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

The topology generated by a basis $\mathcal{B}$ consists of exactly the unions of the elements of $\mathcal{B}$ .

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

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.