Compactness of topological space

Hey.

My assignment says the following:

Let be a topolical space with as its topology. Let be a point not in . Let .

Let is a closed, compact subset of .

(1) Prove that is a topology on . (I have already done this)

(2) SHow that is compact.

I dont know how to show number 2? Could anyone give me a hint or some advice on how to approach this?

Thanks a million.

/Morten

Re: Compactness of topological space

Take an open over of , say . For some , we have , hence necessarily is of the form where is a closed compact subset of . is covered by hence you can extract a finite subcover and you are done (just add ).