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Let X be a metric space with metric d. Let r > 0 and define the open ball center at IE X with radius r>0 by B,(r) = {y € X: d(x,y) < r}. Let T= {BC X: (Vro € B), (3r, > 0), such that B (ro) C B}. 1. Show that r is a topology in X.

Let X be a metric space with metric d. Let r > 0 and define the open ball center at IE X with radius r>0 by B,(r) = {y € X: d(x,y) < r}. Let T= {BC X: (Vro € B), (3r, > 0), such that B (ro) C B}. 1. Show that r is a topology in 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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Let X be a metric space with metric d. Let r > 0 and define the open ball center at IEX with radius r>0 by B,(r) (ye X: d(r, u) <r). Let T= {BC X : (Vro € B), (3r, > 0), such that B(ro) C B}. 1. Show that r is a topology in X. 2. Show that B,(x) is open set with respeet to 7 for all r>0. 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x, 4) Sr} Show that B,(r) is closed set with respect to T. 4. Show that every open set with respect to 7 is a union of open ball(s). A point p is a limit point of E if for all r> 0, there is q # p such that qE EnB,(p). For ECX, define E' be the set of all limit point of E. Show that E is 5. closed if and only if E = EUE'.
Transcribed Image Text:Let X be a metric space with metric d. Let r > 0 and define the open ball center at IEX with radius r>0 by B,(r) (ye X: d(r, u) <r). Let T= {BC X : (Vro € B), (3r, > 0), such that B(ro) C B}. 1. Show that r is a topology in X. 2. Show that B,(x) is open set with respeet to 7 for all r>0. 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x, 4) Sr} Show that B,(r) is closed set with respect to T. 4. Show that every open set with respect to 7 is a union of open ball(s). A point p is a limit point of E if for all r> 0, there is q # p such that qE EnB,(p). For ECX, define E' be the set of all limit point of E. Show that E is 5. closed if and only if E = EUE'.
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