Q1\a\ Let X be an arbitrary infinite set and let the family of all subsets F of X which do not contain a particular point x, EX and the complements FC of all finite subsets F of X show that (X,T) is a topology. b\ The nbhd system N(x) at x in a topological space X has the following properties NO- N(x) for any x E X N1- If NE N(x) then x E N N2- If NE N(x), NC M then M = N(x) N3- If NEN(x), M = N(x) then NO MEN(x) T-AEX, 4*, *S] N4- If NEN(x) then 3M EN(x) such that MCN then MEN(y) for any yЄM Show that there exist a unique topology τ on X. Q2\a\let (X,T) be the topology space and ẞET show that ẞ is base for a topology on X iff for any G open set,x E G then there exist A = ẞ such that x E A ≤ G. b\Let ẞ is a collection of open sets in X show that ẞ is base for a topology on X iff for each x EX the collection ẞx = {BEẞ \xЄ B} is is a nbhd base at x. Q31 Choose only two: al Let A be a subspace of a space X show that FSA is closed iff

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