100 Chapter^2
9. If A is a bounded set in (S, d), show that A is also a bounded set.
10. Show that A C Sis nowhere dense in S if Ext A is everywhere dense
in S.2.10 CONNECTED SETS
By Definition 2.5, two sets A and B are disjoint if An B = 0. We now
introduce the following definition.
Definition 2.29 Let (S, d) be a metric space and let A and B be subsets
of S. Such sets A and B are said to be separated if neither set has a point
in common with the closure of the other, that is, if
AnB=0 and AnB=0
It is clear that if two sets are separated they are disjoint sets sinceAnBcAnB=0
However, two disjoint sets are not necessarily separated. For instance, on
~^1 the sets A = {x: x::::; 1} and B = {x: x > 1} are disjoint, yet not
separated.
Theorem 2. 7 The following properties hold:
1. If d(A, B) > 0, the sets A and B are separated. This condition,
although sufficient, is not necessary.2. If AC C, BCD and C and Dare separated, A and Bare separated.
- Two open sets are separated iff they are disjoint.
- Two closed sets are separated iff they are disjoint.
5. If A is open and A = B U C, where B and C are separated, then B
and C are open.6. If A is closed and A= BU C, where B and C are separated, then B
and C are closed.Proof We shall prove properties 1 and 6, leaving the proofs of the other
parts to the reader.
(1) Let d(A, B) = r > 0. By Definition 2.26,
r =inf{d(a,b): a E A,b EB}and it follows that d( a, b) 2:: r > 0 for every pair of points a E A and b E B.
Hence no point a E A is a contact point of B, since the neighborhood
Ni; 2r (a) does not contain any point of B. Thus An B = 0. Similarly, we
conclude that An B = 0.