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Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f: X→ X x X, x→ (x,x). Its image f(X) is the diagonal Ax = {(x,x) | x ≤ X}.} b) Show that f is a homeomorphism on its image f(X) = Ax, i.e. the restriction f : X→ Ax is a homeomorphism.

Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f: X→ X x X, x→ (x,x). Its image f(X) is the diagonal Ax = {(x,x) | x ≤ X}.} b) Show that f is a homeomorphism on its image f(X) = Ax, i.e. the restriction f : X→ Ax is a homeomorphism.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Certainly! Here is a transcription of the visible text from the image: --- **Exercise 2.** Let \( X \) be a topological space. Endow \( X \times X \) with the product topology. Consider the map \( f : X \to X \times X, x \mapsto (x, x) \). Its image \( f(X) \) is the diagonal \(\Delta_X = \{(x, x) \mid x \in X\}\). b) Show that \( f \) is a homeomorphism on its image \( f(X) = \Delta_X \), i.e. the restriction \( f : X \to \Delta_X \) is a homeomorphism. --- If you have any further questions or need additional explanations, feel free to ask!
Transcribed Image Text:Certainly! Here is a transcription of the visible text from the image: --- **Exercise 2.** Let \( X \) be a topological space. Endow \( X \times X \) with the product topology. Consider the map \( f : X \to X \times X, x \mapsto (x, x) \). Its image \( f(X) \) is the diagonal \(\Delta_X = \{(x, x) \mid x \in X\}\). b) Show that \( f \) is a homeomorphism on its image \( f(X) = \Delta_X \), i.e. the restriction \( f : X \to \Delta_X \) is a homeomorphism. --- If you have any further questions or need additional explanations, feel free to ask!
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