# hw3sol - 147 HW3 Solutions 17.6 Let A B and A denote...

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147 HW3 Solutions 17.6) Let A , B and A α denote subsets of a space X . Prove the following: (a) If A B , then A B . Solution: Let C be some closed subset of X . If B C then A C so by the deﬁnition of U as the intersection of all closed subsets of X containing U , we have that A B . (b) A B = A B . Solution: A B C if and only if A C and B C . Using part (a) we have A A B B , and thus A B A B . But A B A B , and the latter set is closed (ﬁnite unions of closed sets are closed), so A B A B by the deﬁnition of A B . (c) S A α S A α ; give an example of where equality fails. Solution: If S A α C , then for any α , A α C . Thus S A α A α for every α , giving us that S A α S A α . Note: only ﬁnite unions of closed sets are guaranteed to be closed. For an example of equality failing, consider A n = ( 1 n , 1) R . S n Z + A n = (0 , 1) = [0 , 1], but S n Z + A n = S n Z + [ 1 n , 1] = (0 , 1]. 17.8) Let A , B and A α denote subsets of a space X . Determine whether the following equations hold; if an equality fails, determine whether one of the inclusions

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hw3sol - 147 HW3 Solutions 17.6 Let A B and A denote...

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