# 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)

###### 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.

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 |