#1. Let X be a topological space, and let A C X. Suppose that for each x A there is an open set U containing a such that UC A. Show that A is open in X. #2. Let X be a topological space. Suppose that C is a collection of open sets of X such that for every open set U in X and each z EU, there is an element CEC such that x ECCU. Show that C is a basis for the topology of X. #3. Show that the collection T₁:= {U CX:U = Ø or X-U is finite} is a topology on the set X. Is the collection a topology on X? Tx := {U CX : U=Ø or X, or X- U is infinite}/n#4. Let {T} be a family of topologies on X. (a) Show that Ta is a topology on X. (b) Is UTa a topology on X? (c) Show that there is a unique smallest topology on X containing all of the collections Ta (d) Show that there is a unique largest topology on X contained in all of the collections T-

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