Question concerning compact subtopologies on Hausdorf spaces

[Kalimaa23]

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.

 

[Landau]

What about the trivial topology (= only empty and X are open)?

 

[Kalimaa23]

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.