Unformatted text preview: U α ∈ T α : g1 ( f1 α U α ) ∈ T Y ⇔ ∀ α ∈ J ∀ U α ∈ T α : ( f α ◦ g )1 U α ∈ T Y ⇔ ∀ α ∈ J : f α ◦ g : Y → X α is continuous ⇔ f ◦ g : Y → Y X α is continuous where T Y is the topology on Y and T α the topology on X α . (d) . Consider ﬁrst a single map f : A → X , and give A the initial topology so that the open sets in A are the sets of the form f1 U for U open in X . Then f : A → f ( A ) is always continuous [Thm 18.2] and open because f ( A ) ∩ U = f ( f1 U ) for all (open) subsets U of X . Next, note that the initial topology for the set maps { f α  α ∈ J } is the initial topology for the single map f = ( f α ): A → Q X α . As just observed, f : A → f ( A ) is continuous and open. Example : The product topology on Q X α is the initial topology for the set of projections π α : Q X α → X α . References 1...
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 Fall '08
 Brown
 Topology, Continuous function, Empty set, Metric space, Open set, Topological space

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