Countably Compact Space is Weakly Countably Compact

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Theorem

Every countably compact space is a weakly countably compact space.


Proof

Let $T = \struct {S, \tau}$ be a countably compact space.


By definition:

$T$ is weakly countably compact if and only if every infinite subset of $S$ has a limit point in $S$.


By definition of countably compact space:

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


By definition, an $\omega$-accumulation point is a limit point.


Hence the result.

$\blacksquare$


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