Cofinite topology

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Definition

Suppose X is a set. The cofinite topology on X is a topological space structure on X 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 X.
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 X 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.