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Let X be a metric space with metric d. Let r > 0 and define the open ball center at x € X with radius r > 0 by B,(x) = {y € X : d(x,y) < r}. Let T = {BC X : (Vxo € B), (3r70 > 0), such that B, (xo) C B}. A. Show that t is a topology in X. B. Show that B,(x) is open set with respect to r for all r > 0. C.1 Let r > 0 and define the closed ball by B,(x) = {y € X : d(x, y)

Let X be a metric space with metric d. Let r > 0 and define the open ball center at x € X with radius r > 0 by B,(x) = {y € X : d(x,y) < r}. Let T = {BC X : (Vxo € B), (3r70 > 0), such that B, (xo) C B}. A. Show that t is a topology in X. B. Show that B,(x) is open set with respect to r for all r > 0. C.1 Let r > 0 and define the closed ball by B,(x) = {y € X : d(x, y)

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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show complete solution for C.1 and C.2
ΕΧE 11.1 Let X be a metric space with metric d. Let r > 0 and define the open ball center at x € X with radius r > 0 by B,(x) = {y € X : d(x, y) < r}. Let T = {BC X : (Vxo E B), (3r20 > 0), such that B,r, (xo) C B}. A. Show that ī is a topology in X. B. Show that B,(x) is open set with respect to t for all r > 0. C.1 Let r > 0 and define the closed ball by B,(x) = {y E X : d(x,y) < r} Show that B,(x) is closed set with respect to t. and C.2 Show that every open set with respect to t is a union of open ball(s).
Transcribed Image Text:ΕΧE 11.1 Let X be a metric space with metric d. Let r > 0 and define the open ball center at x € X with radius r > 0 by B,(x) = {y € X : d(x, y) < r}. Let T = {BC X : (Vxo E B), (3r20 > 0), such that B,r, (xo) C B}. A. Show that ī is a topology in X. B. Show that B,(x) is open set with respect to t for all r > 0. C.1 Let r > 0 and define the closed ball by B,(x) = {y E X : d(x,y) < r} Show that B,(x) is closed set with respect to t. and C.2 Show that every open set with respect to t is a union of open ball(s).
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