Unformatted text preview: 4. Theorem. Let ( X,d x ) and ( Y,d y ) be metric spaces. For any nonempty set C ⊆ X , if f : C → Y is continuous and C is compact then f ( C ) is compact. 5. Theorem. Let ( X,d x ) and ( Y,d y ) be metric spaces and suppose f : X → Y and g : X → Y are both continuous. Let E ⊆ X be dense in X . (a) If { y t } is a sequence in Y and y t converges both to y * and to ˆ y then y * = ˆ y . (b) If f = g on E then f = g on X . (c) f ( E ) is dense in f ( X ) ....
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 Fall '08
 JohnNachbar
 Economics, Topology, Metric space, Topological space, John Nachbar Fall, John Nachbar

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