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et X be a metric space with metric d. Let r > 0 and define the open ball center at EX with radius r >0 by B,(x) = {y € X : d(x,y) < r}. et {BCX:(Vro E B), (3r10 > 0), such that B, (ro) C B}. Show that T is a topology in X and Show that B,(x) is open set with respect to T for allr>0

et X be a metric space with metric d. Let r > 0 and define the open ball center at EX with radius r >0 by B,(x) = {y € X : d(x,y) < r}. et {BCX:(Vro E B), (3r10 > 0), such that B, (ro) C B}. Show that T is a topology in X and Show that B,(x) is open set with respect to T for allr>0

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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EXE 1.1 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 = {BCX: (Vro e B), (3rzo > 0), such that B (ro) C B}. A. Show that T is a topology in X and Show that B,(x) is open set with respect to T for all r > 0
Transcribed Image Text:EXE 1.1 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 = {BCX: (Vro e B), (3rzo > 0), such that B (ro) C B}. A. Show that T is a topology in X and Show that B,(x) is open set with respect to T for all r > 0
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