Posted by J. Holmer on September 30, 2007, 3:52 pm, in reply to "HW5 1(a)" (1) B \subset T is a subbase if the smallest topology containing B is T (the smallest topology is the intersection of all topologies containing T; we also say "the topology generated by B" in place of "the smallest topology containing B" ) (2) B \subset T is a subbase if for every U in T and x_0 in U, there exists B_1, ... , B_N in B such that x_0 \in (B_1\cap . . .\cap B_N) \subset U. We wish to prove that (1) iff (2). To do this, we'll first prove Lemma. Let A be a collection of subsets of X. Then the smallest topology (on X) containing A is equal to the collection A' defined as follows: A' is the collection of all subsets of X that can be written as the (abitrary) union of finite intersections of sets from A. To prove this lemma, one just needs to make two observations. The collection A' is a topology (this follows from set theory facts describing the manner in which intersection commutes with union) Any topology that contains A contains A' (this is clear by definition of topology). Now, using the lemma, do you see how (1) implies (2) and conversely?
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X= a topological space with topology T. The two definitions are 134
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