PlanetMath: cofinite and cocountable topologies

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cofinite and cocountable topologies

(Definition)

The cofinite topology on a set is defined to be the topology $\mathcal{T}$ where

$\displaystyle \mathcal{T} = \{A \subseteq X \mid X \setminus A \hbox{ is finite, or } A=\varnothing \}.$

In other words, the closed sets in the cofinite topology are

and the finite subsets of

Analogously, the cocountable topology on is defined to be the topology in which the closed sets are and the countable subsets of .

The cofinite topology on is the coarsest topology on .

The cofinite topology on a finite set is the discrete topology. Similarly, the cocountable topology on a countable set is the discrete topology.

A set together with the cofinite topology forms a compact topological space.

"cofinite and cocountable topologies" is owned by yark. [ full author list (2) | owner history (1) ]

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