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| A topological space $X$ is said to be \emph{weakly countably compact} |
A topological space $X$ is said to be \emph{limit point compact} if every infinite subset of $X$ has a limit point. |
| (or \emph{limit point compact}) |
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| if every infinite subset of $X$ has a limit point. |
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| Every countably compact space is weakly countably compact. |
Limit point compactness is equivalent to countable compactness if $X$ is $T_1$ and is equivalent to \PMlinkname{compactness}{Compact} if $X$ is a metric space. |
| The converse is true in \PMlinkname{$\mathrm{T}_1$ spaces}{T1Space}. |
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| A metric space is weakly countably compact if and only if it is compact. |
\PMlinkescapeword{compact} |
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| An easy example of a space $X$ |
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| that is not weakly countably compact |
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| is any infinite set with the discrete topology. |
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| A more interesting example is the countable complement topology |
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| on an uncountable set. |
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