Question concerning compact subtopologies on Hausdorf spaces

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The discussion centers on the existence of a coarser topology on a Hausdorff space (X,T) such that closed subsets are also compact. It is established that any compact subset of a Hausdorff space is closed. The trivial topology, which includes only the empty set and the entire space X as open sets, is proposed as a solution. This topology satisfies the requirement since it inherently makes all closed sets compact.

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Kalimaa23
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Greetings,

I'm helping out a student with her upcoming topology exam and something has be stomped. It's probably simple but I'm not seeing it at the moment.

Consider a Hausdorf space (X,T). Any compact subset of X is therefore closed.

The question is to prove the existence of a coarser topology on (X,T) so that closed also implies compactness. I'm basically trying to find a coarser topology on X that makes it compact.

Thanks in advance.
 
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What about the trivial topology (= only empty and X are open)?
 
That had occurred to me, but I though there might be something you could prove about a topology finer than the trivial one. But I guess it fits the bill.
 

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