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Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f: X→ Xx X,x+ (x,x). Its image f(X) is the diagonal Ax = {(x,x) | x = X}.} a) Show that f is continuous.

Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f: X→ Xx X,x+ (x,x). Its image f(X) is the diagonal Ax = {(x,x) | x = X}.} a) Show that f is continuous.

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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Exercise 2. Let \( X \) be a topological space. Endow \( X \times X \) with the product topology. Consider the map \( f: X \rightarrow X \times X, x \mapsto (x, x) \). Its image \( f(X) \) is the diagonal \(\Delta_X = \{(x, x) | x \in X\} \). a) Show that \( f \) is continuous.
Transcribed Image Text:Exercise 2. Let \( X \) be a topological space. Endow \( X \times X \) with the product topology. Consider the map \( f: X \rightarrow X \times X, x \mapsto (x, x) \). Its image \( f(X) \) is the diagonal \(\Delta_X = \{(x, x) | x \in X\} \). a) Show that \( f \) is continuous.
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