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Suppose that z B y, 1
n B y, 1
. We observe that
d x, z B x, . d x, y d y, z 1 .
3 14. Given a point x in a metric space X and given r 0, prove that
B x, r
B x, r .
Give an example to show that the latter inequality does not have to be an equation.
We know from an earlier theorem that the set B x, r is a closed set that includes B x, r .
15. Prove that if x R k and r 0 then
B x, r B x, r and 124 I x, r I x, r .
Both assertions follow automatically from Exercise 14.
16. Which of the following pairs of sets are separated from each other in the metric space R? a. 0, 1 and 2, 3 . Yes. b. 0, 1 and 1, 2 . Yes. c. 0, 1 and 1, 2 . No because 0, 1 d. Q and R 1, 2 1 . Q. No. 17. Prove that if A and B are closed in a metric space X and disjoint from one another then A and B are separated
from each other.
Suppose that A and B are closed and disjoint from one another. Since A A and B B, the fact
that A B A B follows at once from the fact that A B .
18. Prove that if A and B are open in a metric space X and disjoint from one another then A and B are separated
from each other.
Suppose that A and B are open and disjoint from one another. Given any number x A, we deduce
from the fact that A is a neighborhood of x and A B that x is not close to B. Therefore
A B and we see similarly that A B .
19. Suppose that S is a subset of a metric space X. Prove that the two sets S and X S will be separated from
each other if and only if the set S is both open and closed. What then do we know about the subset S of the
metric space R for which S and R S are separated from each other?
Suppose that S and X S are separated from each other. To show that S is open, suppose that
x S. Since S
X S we know that x is not close to X S. Choose 0 such that
x , x
and observe that x , x
S. Thus S is open and a similar argument shows that X S is also
open. We therefore know that if the sets S and X S are separated from one another then S is both
open and closed.
Now suppose that S is both open and closed. Since the two set S and X S are closed and disjoint
from one other they are separated from one another.
20. Prove that a metric space X is connected if and only if it cannot be written as the union of two nonempty sets
that are separated from one another.
This assertion follows at once from Exercise 19.
21. Prove that if S is a connected subspace of a metric space X then the subspace S is also connected.
We want to show that if the space S fails to be connected then the space S must also fail to be
Assume that the space S fails to be connected and choose a nonempty open closed subset E of
the space S such that E S. In order to show that S fails to be connected we shall show that both
S E and S E are nonempty open subsets of the space S.
Using the fact that E is open in S and the theorem on open subsets of a subspace, choose a set V
that is open in the metric space X such that E S V. Since S E S V the set...
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