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.
- 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.