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Exercise 4. Suppose f: I→ X is a continuous map from the interval ICR to the topological space X. Consider the graph Graph(f) C IX X of f, defined as usual by Graph(f) = {(t, f(t)) | te I} CRX X}. ph/ the c) Assume X is Hausdorff. Show that the closure Graph(f) is compact if and only if I is a bounded interval and the closure f(I) in X is compact.

Exercise 4. Suppose f: I→ X is a continuous map from the interval ICR to the topological space X. Consider the graph Graph(f) C IX X of f, defined as usual by Graph(f) = {(t, f(t)) | te I} CRX X}. ph/ the c) Assume X is Hausdorff. Show that the closure Graph(f) is compact if and only if I is a bounded interval and the closure f(I) in X is compact.

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 4. Suppose f: I→ X is a continuous map from the interval ICR to the topological space X. Consider the graph Graph(f) CIXX of f, defined as usual by Graph(f) = {(t, f(t)) |te I} CRX X}. nh/c by snow c) Assume X is Hausdorff. Show that the closure Graph(f) is compact if and only if I is a bounded interval and the closure f(I) in X is compact.
Transcribed Image Text:Exercise 4. Suppose f: I→ X is a continuous map from the interval ICR to the topological space X. Consider the graph Graph(f) CIXX of f, defined as usual by Graph(f) = {(t, f(t)) |te I} CRX X}. nh/c by snow c) Assume X is Hausdorff. Show that the closure Graph(f) is compact if and only if I is a bounded interval and the closure f(I) in X is compact.
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