1. Let X be a topological space and AC X. Then prove that 2/P xA if and only if every open set U containing a intersects A. (4) if B is a basis for X, then xe A if and only if every basis element BEB containing intersects A. Prove the following: (a) Let X be an infinite set. Then check whether X with cofinite topology satisfy T and T axioms or not. Let X be a compact topological space and Y be a closed subspace of X. Then Y is compact. 12 2. Let X be a topological space and Y CX be a subspace of X. Then prove that (a) Y is compact if and only if every covering of Y by open sets in X has a subcovering that contains Y. If Y is compact and X is Hausdorff, then Y is closed. 4. Let X be a topological space and Y C X is a connected subspace of X. Then prove that (a) if A and B form a separation of X, then Y lies entirely within either A or B. if bf Y CZCY, then Z is also connected. WHY