On the topology of pointwise convergence on the ... - CARMA
On the topology of pointwise convergence on the ... - CARMA
On the topology of pointwise convergence on the ... - CARMA
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⋃n∈N (x n+εB X ). For each n ∈ N, let C n := f −1 (x n +εB X ).Since each x n + εB X is closed in <str<strong>on</strong>g>the</str<strong>on</strong>g> σ(X, B) <str<strong>on</strong>g>topology</str<strong>on</strong>g> eachset C n is closed in A and moreover, W ⊆ ⋃ n∈N C n. Since Wis <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> sec<strong>on</strong>d Baire category in A <str<strong>on</strong>g>the</str<strong>on</strong>g>re exist a n<strong>on</strong>emptyopen set U ⊆ W and a k ∈ N such that U ⊆ C k . ThenU ⊆ O ε ∩ W and so O ε is indeed dense in A. Clearly, f is‖ · ‖-c<strong>on</strong>tinuous at each point <str<strong>on</strong>g>of</str<strong>on</strong>g> ⋂ n∈N O 1/n.❦ ✂✁Theorem 4 Suppose that A is a topological space with countabletightness that possesses a rich family F <str<strong>on</strong>g>of</str<strong>on</strong>g> Baire subspacesand suppose that X is an L 1 -predual. Then for anyboundary B <str<strong>on</strong>g>of</str<strong>on</strong>g> X and any c<strong>on</strong>tinuous functi<strong>on</strong> f : A →(X, σ(X, B)) <str<strong>on</strong>g>the</str<strong>on</strong>g>re exists a dense subset D <str<strong>on</strong>g>of</str<strong>on</strong>g> A such that fis c<strong>on</strong>tinuous with respect to <str<strong>on</strong>g>the</str<strong>on</strong>g> norm <str<strong>on</strong>g>topology</str<strong>on</strong>g> <strong>on</strong> X at eachpoint <str<strong>on</strong>g>of</str<strong>on</strong>g> D provided X is ℵ 0 -m<strong>on</strong>olithic in <str<strong>on</strong>g>the</str<strong>on</strong>g> σ(X, Ext(B X ∗))<str<strong>on</strong>g>topology</str<strong>on</strong>g>.