I know that if $X$ is a Hausdorff topological space and $A$ is compact in $X$, then $A$ is closed in $X$. My question is that if $A$ is a closed set in $X$ (where $X$ is Hausdorff), what extra condition is needed to ensure that $A$ is compact in $X$?
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There’s really not much that can be said in general: it depends very much on the Hausdorff space $X$. Some examples:
Brian has already given you many examples explaining that compactness of $A$ strongly depends on properties of the Hausdorff space $X$. It is worth to add to his list the following one:
This is one of the most important characterisations of compact subsets of metric spaces.
Many interesting properties and other characterisations you can find in the Wikipedia article: http://en.wikipedia.org/wiki/Compact_space