Could you explain how to show 7.11 in detail? Definition. Let X and Y be topological spaces. A function or map f : X → Y is acontinuous function or continuous map if and only if for every open set U in Y,f-l(U) is open in X.Definition. Let f : X → Y be a function between topological spaces X and Y, andlet x e X. Then f is continuous at the point x if and only if for every open setV containing f(x), there is an open set U containing x such that f(U) c V. Thus afunction f : X→ Y is continuous if and only if it is continuous at each point.Theorem 7.9. If ƒ : X → Y and g : Y → Z are continuous, then their compositiongof : X → Z is continuous.We can paste two continuous functions together if the pieces are either both closedor both open.AU B, where A, B are closed in X. Let f :Theorem 7.10 (Pasting lemma). Let X =A → Y and g : B →h : AUB → Y defined by h = f on A and hY be continuous functions that agree on An B. Then the functiong on B is continuous.Theorem 7.11 (Pasting lemma). Let X = A U B, where A, B are open in X. Let f :A → Y and g : B → Y be continuous functions which agree on An B. Then the functionh : AUB-Y defined by h = ƒ on A and h= g on B is continuous.

Name: Could you explain how to show 7.11 in detail? Definition. Let X and Y
Uploaded: 2024-03-20T07:06:16.000Z
Channel: Bartleby
Description: Could you explain how to show 7.11 in detail? Definition. Let X and Y be topological spaces. A function or map f : X → Y is acontinuous function or continuous map if and only if for every open set U in Y,f-l(U) is open in X.Definition. Let f : X → Y