basis (topology)

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

Equivalently, is a basis if and only if whenever U is open and xU then there is an open set V such that xVU.

The topologyMathworldPlanetmath generated by a basis consists of exactly the unions of the elements of .

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


A collectionMathworldPlanetmath of subsets B of X is a basis for some topology on X if and only if each xX is in some element BB and whenever B1,B2B and xB1B2 then there is B3B such that xB3B1B2.

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 endpointsMathworldPlanetmath 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 intersectionMathworldPlanetmath of two intervals of length 1).

2. More generally, the set of open ballsPlanetmathPlanetmath 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.

Title basis (topology)
Canonical name Basistopology
Date of creation 2013-03-22 12:05:03
Last modified on 2013-03-22 12:05:03
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 16
Author rspuzio (6075)
Entry type Definition
Classification msc 54A99
Synonym basis
Synonym base
Synonym topology generated by a basis
Related topic Subbasis
Related topic CompactMetricSpacesAreSecondCountable