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d(r, y) 1+d(x,y) 5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) = Show that f is a metric on X.

d(r, y) 1+d(x,y) 5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) = Show that f is a metric on X.

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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I need help with #5.

1. Let E and F be compact sets in a metric space (X, d). Show that EUF is compact using the definition of compactness. 2. Let (X, d) be a metric space and EC X. Prove that if E is compact, then E is bounded. 3. Suppose E C R. With respect to the Euclidean metric, is it possible that E' :neN. If so, give an example; otherwise explain. 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed. d(r, y) Show 1+ d(x,y)' 5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) = that f is a metric on X.
Transcribed Image Text:1. Let E and F be compact sets in a metric space (X, d). Show that EUF is compact using the definition of compactness. 2. Let (X, d) be a metric space and EC X. Prove that if E is compact, then E is bounded. 3. Suppose E C R. With respect to the Euclidean metric, is it possible that E' :neN. If so, give an example; otherwise explain. 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed. d(r, y) Show 1+ d(x,y)' 5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) = that f is a metric on X.
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