Closed subset of a compact space
= Closed subset of a compact space
{title2=$F\text{ closed in compact }X\ \Longrightarrow\ F\text{ compact}$}
Every <closed subset> $F$ of a <compact space> $X$ is <compact> in its <subspace topology>. An <open cover> of $F$ by ambient <open sets>, together with $X\setminus F$, covers $X$. A <finite subcover> for $X$ still covers $F$ after the extra member is discarded. The result does not require $X$ to be <Hausdorff>.