Let T be the cofinite topology for R Is 0 infinity compact P

Let T be the co-finite topology for R. Is [0, infinity) compact? Prove your answer.

Solution

Let X= R=[0,infinity),    Let T be the co-finite topolgy in R. Now , we want to prove that (X,T) is compact space.

Proof:-    Aim to prove that Any infinite set with finite complement topology is compact. Let X be the infinite set with co-finite topolgy.Let {U_a} be covering of X. then X-U_a is a finite set as co-finiteness. say {x_1,x_2,....x_n}.LetU_ai be one of the open sets that contains x_i Then {U_a1,U_a2,....U_an} = X. so, Every open covering has a finite subcollection. Therefore X is compact