Has trouble understanding finite set while I was solving theorem on topological space which is
Every co-finite topology is compact. Let (X,T) be topological space. In co-finite topology X is infinite set. And every open set G of topology is infinite because x-G is finite( as per definition of co-finite topology).
Now since open covering contain open set of topology hence must contain subset of topology T which are infinite.
We know topological space is compact if every open cover of Set X has finite subcover.
Does by finite subcover it means Set of finite number of elements as set which are further containing infinite elements (subsets which are infinite set) as in case of co-finite topology.
E.g A= {{10,11,12....... Infinity},{2n; n is natural number}, {n+2; n is natural number} } Here A has three elements( taking them as a1, a2 and a3) and all three are infinite set. So A must be finite but a1,a2 and a3 are infinite.