Definition:Countably Compact Space

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Definition

Definition 1

A topological space $T = \left({S, \tau}\right)$ is countably compact if and only if:

every countable open cover of $T$ has a finite subcover.


Definition 2

A topological space $T = \left({S, \tau}\right)$ is countably compact if and only if:

every countable set of closed sets of $T$ whose intersection is empty has a finite subset whose intersection is empty.

That is, $T$ satisfies the countable finite intersection axiom.


Definition 3

A topological space $T = \left({S, \tau}\right)$ is countably compact if and only if:

every infinite sequence in $S$ has an accumulation point in $S$.


Definition 4

A topological space $T = \left({S, \tau}\right)$ is countably compact if and only if:

every countably infinite subset of $S$ has an $\omega$-accumulation point in $S$.


Definition 5

A topological space $T = \left({S, \tau}\right)$ is countably compact if and only if:

every infinite subset of $S$ has an $\omega$-accumulation point in $S$.


Also see

  • Results about countably compact spaces can be found here.