Could you explain how to show 7.7 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-(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 7.7. LetX be a space with a dense set D, and let Y be Hausdorff. Let f:X → Yand g : X → Y be continuous functions such that for every d in D, f(d)for all x in X, f(x) = g(x).g(d). Then

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Description: Could you explain how to show 7.7 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-(U) is open in X.Definition. Let f : X → Y be

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Theorem 7.7. LetX be a space with a dense set D, and let Y be Hausdorff. Let f : X → Y and g : X → Y be continuous functions such that for every d in D, f(d) = g(d). Then for all x in X, f(x) = g(x).

Theorem 7.7. LetX be a space with a dense set D, and let Y be Hausdorff. Let f : X → Y and g : X → Y be continuous functions such that for every d in D, f(d) = g(d). Then for all x in X, f(x) = g(x).

Elements Of Modern Algebra

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ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.3: Properties Of Composite Mappings (optional)

Problem 6E: 6. a. Give an example of mappings and , different from those in Example , where is one-to-one, is...

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Could you explain how to show 7.7 in detail?

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 → Y be a function between topological spaces X and Y, and
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 is continuous if and only if it is continuous at each point.
Theorem 7.7. LetX be a space with a dense set D, and let Y be Hausdorff. Let f:X → Y
and g : X → Y be continuous functions such that for every d in D, f(d)
for all x in X, f(x) = g(x).
g(d). Then

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