Definition:Countably Compact Space
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Definition
Definition 1
A topological space $T = \struct {S, \tau}$ is countably compact if and only if:
- every countable open cover of $T$ has a finite subcover.
Definition 2
A topological space $T = \struct {S, \tau}$ 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 $T = \struct {S, \tau}$ 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 $T = \struct {S, \tau}$ 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.