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
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ISBN:9780470458365
Author:Erwin Kreyszig
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Exercise 4 Need part a, b, and c
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)) | t € I} < R × X}.
a) Show that I is homeomorphic to Graph(f), with the topology on Graph(f) induced from the
product topology on RX X.
b) Show that both Graph (f) and its closure Graph(f) in R × 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.
1
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) C IX X of f, defined as usual by Graph(f) = {(t, f(t)) | t € I} < R × X}. a) Show that I is homeomorphic to Graph(f), with the topology on Graph(f) induced from the product topology on RX X. b) Show that both Graph (f) and its closure Graph(f) in R × 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. 1
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