Topology of Plane Sets of Points 99
Thus the diameter of A is the supremum (least upper bound) of the set of
all distances between pair of points of A.
Definition 2.28 A set A C S is said to be bounded iff 0 :::; ~(A) <
+oo. Alternatively, a set A is bounded iff it is contained in an open ball
Nr(xo),xo E S, for some r > 0.
A set that is not bounded is said to be unbounded. A set A C S may or
may not be bounded depending on the metric chosen in S. For instance,
(C, x) is bounded, whereas (C, d) is unbounded.
The empty set 0 is supposed to be bounded with diameter 0.
Theorem 2.6 The following properties hold:
- A= {x: x E S,d(x,A) = O}.
- A is everywhere dense in S iff d( x, A) = 0 for every x E S.
EXERCISES 2.4
- Apply Definition 2.12 to show that in (C, d) N6(a) is an open set, and
Definition 2.13 to show that Ns[a] is a closed set.
- Let (S, d) be a metric space and let x and y be distinct points of S.
(a) Show that there are 8-neighborhoods N6 1 (x) and N6 2 (y), with 81 >
0,8 2 > 0, such that N6 1 (x) n N6 2 (y) = 0 (this is called the separation
property). (b) Let y E N6(x). Show that there is a N6•(y) (8^1 > 0)
such that N61(y) C N6(x).
- Let (S, d) be a metric space and A = {x 1 , x 2 , ••• , xn} a finite subset
of S. Prove that A' is closed.
4. Consider in (C,d) the set M = {z: Rez E Q,Imz E Q} where Q de-
notes the set of rational numbers. Find: (a) IntM; (b) M; (c) M;
(d) oM.
- Prove the properties stated in Theorems 2.2 to 2.5.
6. Let (S, d) be a metric space, and let A, BC S. Prove the following.
(a) A= A
(c) (A
1
)^1 =Int A
(e)AnBcAn.B
(b) (A)'= Int A'
(d) AUB=AUB
(f) [(.B)']' = IntB
7. Let (S, d) be a metric space, and let A, BC S. Prove the following.
(a) Int AU lntB c Int(A U B)
(b) Int An IntB = Int(A n B)
(c) Ext An ExtB = Ext(A U B)
(d)oA=AnA' (e)oA=oA.
- A set A C S, where (S, d) is a metric space, is said to be discrete if
all its points are isolated. Consider the space ( C, d) and a discrete set
. A C C. Prove that A is countable.
