(a) Let X = {a}. Explain why the only topology on X is {Ø, X}.
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Transcribed Image Text:(a) Let X = {a}. Explain why the only topology on X is {Ø, X}.
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A topology on a singleton set is a collection of subsets of the singleton set which contains the empty set and the entire singleton set and is closed under unions and finite intersections. In this case, the empty set and the entire singleton set are all possible subsets of the singleton set (that is, the subsets of are precisely ∅ and X, and you can easily check that these are indeed closed under union and finite intersection. There is therefore only one possible topology on the singleton set, which is τ={∅,X}
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