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.

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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

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Answered: Theorem 7.21. IfX is normal and f : X →…

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Theorem 7.21. IfX is normal and f : X → Y is continuous, surjective, and closed, then Y is normal.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...

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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 a
continuous function or continuous map if and only if for every open set U in Y,
f-'(U) is open in X.
Definition. Let f : X
let x e X. Then f is continuous at the point x if and only if for every open set
V containing f(x), there is an open set U containing x such that f(U) c V. Thus a
function 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 : X
in X, f(U) is open in Y.
→ Y is open if and only if for every open set U
Theorem 7.21. IfX is normal and f : X
Y is normal.
→ Y is continuous, surjective, and closed, then
Theorem 7.24. Let X be compact, and let Y be Hausdorff. Then any continuous function
f :X → Y is closed.

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