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 $\mathcal{B}$.

Proposition.

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

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 $1$).

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

Title basis (topology) Basistopology 2013-03-22 12:05:03 2013-03-22 12:05:03 rspuzio (6075) rspuzio (6075) 16 rspuzio (6075) Definition msc 54A99 basis base topology generated by a basis Subbasis CompactMetricSpacesAreSecondCountable