Compactness of topological space
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.
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 ).