topological

kingex1124

想問問 為什麼 co-finite topology for a fixed infinite set X.then (X,T) is a T1-space but not a T2-space.

xvmon123

since T is co-finite topology
any open set must containing infinitely many points
recall that the definition of T1 and T2 space
(X,T) is T1 that is clearly
to show (X,T) is not T2
Let a, b be disjoint elements in X
     A, B be open in (X,T) containg a, b respctively
so A^c and B^c are only have finite elements
=>A^c union B^c also have finitely many elements
=>A intersect B =(A^c union B^c)^c have infinitely many elements which is a non-empty set
this shows T2
OR
If A intersect B is empty
=>A contained in B^c
=>A have finitely many elements whch contradict to A must have infinitely many elements

I hope it's useful ^_^