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If {T

_{a}} is a family of topologies on X, show that (intersection)T

_{a}is a topology on X. Is UT

_{a}a topology on X?

Sorry for crappy notation; I don't know my way around the symbols yet.

For the latter, I say "not necessarily"; for example, if X={a,b,c}, T

_{1}={{a}}, and T

_{2}={{b}}, then T

_{1}UT

_{2}={{a},{b}} is not a topology on X because {a,b} is not in the union. If this is wrong, please correct me. However, I am having a very difficult time with these proofs; for the former part of the question, I know the intersection will yield a "coarsest" subset of X, but proving it is a topology is bewildering me.

Thanx for any help.