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First of all, I hope this is an OK place for a topology question. I was debating here or of course set theory, but I guess this is right. Anyway.

I'm studying out of Munkres' book, and I'm looking at a certain problem. The problem stated is:

THEOREM: If A is a basis for a topology on X, then the topology generated by A equals the intersection of all topologies on X that contain A.

Basically, I interpreted this to mean that the basis for a topology on X is necessarily an element of every topology on X, and here's my proof:

PROOF: Suppose to the contrary, that there exists an element in the intersection of all topologies that is not in A. Hence there exists an open set U in X such that for each a in U, there is no basis element A_i such that x is in A_i and A_i is a subset of U. But since this is a topology, x must be in the intersection of U and some element of our basis collection. This would create a basis for x which is a contradiction.

See, I'm really not sure if I did this correctly. My "leap" is where I assume the open set containing U intersects A_i, I think, and that's where I think I go wrong. Could anyone help? Thanks in advance!