Suppose you are given a set X and a collection B of subsets of X. Under...

   

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Suppose you are given a set X and a collection B of subsets of X. Under what circumstances is there a topology for which B is a basis? Note how this question is different from the question that motivated Theorem 3.1. That theorem describes when a collection of sets forms a basis for a given topology. The next theorem describes when a collection of sets forms a basis for some topology on X. Suppose you are given a set X and a collection B of subsets of X. Under what circumstances is there a topology for which B is a basis? Note how this question is different from the question that motivated Theorem 3.1. That theorem describes when a collection of sets forms a basis for a given topology. The next theorem describes when a collection of sets forms a basis for some topology on X.

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Your question relates to the concept of a basis for a topology on a set A basis mathcalB for a topology mathcalT on a set X is a collection of subsets ... View the full answer