1. Let N be a collection of subsets of X, and let O be the smallest...

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1. Let N be a collection of subsets of X, and let O be the smallest topology on X that contains N. a) Show that f: W X is continuous is open in W for every NEN. Hint: Think about the topology P = {U If (U) is open in W}. b) Now let A C X be a subset, and write OA for the subspace topology. Write: N = {ANN M N E N}. Show that OA is the smallest topology on A containing NA. Hint: Use part a) to show that the identity function (of sets) gives continuous maps (A, O) (A, NA) 2. Let X and Y be topological spaces, and suppose UCn, where each Ci is a closed subset of X. Show that a function f: X Y is continuous iff each restriction flc, is continuous. Is it true if the subsets are not closed?

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