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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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