S19 - U α ∈ T α g-1 f-1 α U α ∈ T Y ⇔ ∀ α ∈...

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1st December 2004 Munkres § 19 Ex. 19.7. Any nonempty basis open set in the product topology contains an element from R , cf. Example 7p. 151. Therefore R = R ω in the product topology. ( R is dense [Definition p. 191] in R ω with the product topology.) Let ( x i ) be any point in R ω - R . Put U i = R if x i = 0 R - { 0 } if x i = 0 Then U i is open in the box topology and ( x i ) U i R ω - R . This shows that R is closed so that R = R with the box topology on R ω . See [Ex 20.5] for the closure of R in R ω with the uniform topology. Ex. 19.10. (a) . The topology T (the initial topology for the set maps { f α | α J } ) is the intersection [Ex 13.4] of all topologies on A for which all the maps f α , α J , are continuous. (b) . Since all the functions f α : A X α , α J , are continuous, S = S α ⊂ T . The topology T S generated by S , which is the coarsest topology containing S [Ex 13.5], is therefore also contained in T . On the other hand, T ⊂ T S , for all the functions f α : A X α , α J , are continuous in T S and T is the coarsest topology with this property. Thus T = T S . (c) . Let g : Y A be any map. Then g : Y A is continuous ⇔ ∀ U ∈ S : g - 1 ( U ) ∈ T Y ⇔ ∀
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Unformatted text preview: U α ∈ T α : g-1 ( f-1 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 f-1 U for U open in X . Then f : A → f ( A ) is always continuous [Thm 18.2] and open because f ( A ) ∩ U = f ( f-1 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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