# Cofinite topology

From Topospaces

## Definition

Suppose is a set. The **cofinite topology** on is a topological space structure on that can be defined in the following equivalent ways:

Type of description | Details |
---|---|

in terms of open subsets | The open subsets are precisely the empty set and the cofinite subsets, i.e., the subsets whose complements are finite subsets of . |

in terms of closed subsets | The closed subsets are precisely the whole space and the finite subsets. |

in terms of a subbasis | We can take as a subbasis the complements of singleton subsets in the space. |

as a coarsest topology | The coarsest topology on for which it is a T1 space. |

If the set is finite, the cofinite topology makes it a discrete space.

## Facts

- A set equipped with the cofinite topology is a Toronto space.
- Any bijection between two sets is a homeomorphism between them as topological spaces with the cofinite topology.
- The cofinite topology is the Zariski topology on any connected one-dimensional algebraic variety over a field.
- Any set equipped with the cofinite topology is a Noetherian space, and hence a hereditarily compact space.