How do I show 7.21? Could you explain this in great detail? Thank you! 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-'(U) is open in X.Definition. Let f : Xlet 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 be a function between topological spaces X and Y, and→ Y is continuous if and only if it is continuous at each point.Definition. A function f : X → Y is closed if and only if for every closed set A in X,f(A) is closed in Y. A function f : Xin X, f(U) is open in Y.→ Y is open if and only if for every open set UTheorem 7.21. IfX is normal and f : XY is normal.→ Y is continuous, surjective, and closed, thenTheorem 7.24. Let X be compact, and let Y be Hausdorff. Then any continuous functionf :X → Y is closed.

Name: How do I show 7.21? Could you explain this in great detail? Thank you!
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: How do I show 7.21? Could you explain this in great detail? Thank you! 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-'(U) is open in X.D