TOPOLOGY For a metric space X d boundary of a set A is defin

((TOPOLOGY))

For a metric space (X, d), boundary of a set A is defined as bdA = clA cl(X A). Prove each of the following for a set A:

(i) The bdA is a closed set.

(ii) A is closed if and only if bdA A. (iii) clA = Ao bdA.

Solution

Ans-

On the one hand bd(bd A) = cl(bd A) int(bd A). We claim, however, that bd A is
closed. Indeed, bd A = cl A \\ cl mnA, so it is the intersection of two closed sets and is therefore
closed. Hence bd(bd A) = bd A int(bd A) from which it follows that bd(bd A) bd A.

Solution. We have
jjx
m
sin mjj = j sin mj jjx
jj:
Hence
P
jjx
m
sin mjj
P
jjx
m
jj. Since
P
x
m
m
jj jjx
m
converges absolutely, by the Comparison
Test,
P
jjx
m
sin mjj converges. Since R
is complete and normed, absolute convergence P
x
m
sin m is enough to show its convergence.
n

Solution. We have
jjx
m
sin mjj = j sin mj jjx
jj:
Hence
P
jjx
m
sin mjj
P
jjx
m
jj. Since
P
x
m
m
jj jjx
m
converges absolutely, by the Comparison
Test,
P
jjx
m
sin mjj converges. Since R
is complete and normed, absolute convergence P
x
m
sin m is enough to show its convergence.
n