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) 0). such that B (ro) C B). Show that r is a topology in X. Show that B,(r) is open set with respect to 7 for all r> 0. 3. Let r>0 and define the closed ball by B,(1) = {y €X: d(1, y) Sr} Show that B,(r) is closed set with respect to 7.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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) (ye X: d(r, u) <r).
Let
T= {BCX: (Vfo € B), (Gr, > 0). such that B (ro) C B).
Show that r is a topology in X.
Show that B,(r) is open set with respect to 7 for all r> 0.
3.
Let r>0 and define the closed ball by
B,(1) = {y €X: d(1, y) <r}
%3D
Show that B,(r) is closed set with respect to 7.
Transcribed Image Text: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) (ye X: d(r, u) <r). Let T= {BCX: (Vfo € B), (Gr, > 0). such that B (ro) C B). Show that r is a topology in X. Show that B,(r) is open set with respect to 7 for all r> 0. 3. Let r>0 and define the closed ball by B,(1) = {y €X: d(1, y) <r} %3D Show that B,(r) is closed set with respect to 7.
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