i. PROVE THAT THE EXCLUDED POINT TOPOLOGY ON ANY SET WITH AT LEAST 2 POINTS IS NOT HAUSDORFF. ii. PROVE THAT PARTICULAR POINT TOPOLOGY ON ANY SET WITH AT LEAST 2 POINTS IS NOT HAUSDORFF

A set F is called an Fo-set (or an Fo) if it is a countable union of closed sets. Likewise, a set G is called a Gs-set (or a Gs) if it is a countable intersection of open sets. Then show that (a) [0, 1] is a Gs set. (b) (a,b] is a both Gs and an Fo set. (c) Q is an F set, but not a Gs set.

Consider X = {a1, a2, ..., an}Can a topology of X with 3 open and one with 4 open be equivalent?

Prove that if A is a transitive set then UA is a transitive set.