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}. a) Show that I is homeomorphic to Graph(f), with the topology on Graph(f) induced from the product topology on R X X. b) Show that both Graph(f) and its closure Graph(f) in Rx X are connected. 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}. a) Show that I is homeomorphic to Graph(f), with the topology on Graph(f) induced from the product topology on R X X. b) Show that both Graph(f) and its closure Graph(f) in Rx X are connected. 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
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
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