Topology of Plane Sets of Points 95If N is a neighborhood of a, the set N' = N - {a} is called a deleted
neighborhood of a. In particular, the set
N~(a) = Ns(a) - {a}
is called a deleted 8-neighborhood of a (sometimes a deleted neighborhood
of a for short).Definition 2.10 For every 8 > 0 the set
Ns[a] = {x: x ES, d(x, a)::::; 8}
is called a closed spherical neighborhood, a closed 8-neighborhood, or a closed
ball with center a and radius 8. In the case of a plane set of points the set
Ns [a] is also called a closed disk or a closed circular neighborhood.Definition 2.11 In (<C,d) the set N = {z: lzl > R}, i.e., the exterior
of a circle with center at the origin and arbitrary radius, is called a neigh-
borhood of oo. In (<C*, x) a 8-neighborhood of oo is defined in the usual
manner, namely, N 0 (00) = {z: x(z,oo) < 8}.
In what follows whenever we refer to a 8-neighborhood of a point an
open 8-neighborhood is to be understood.
Definition 2.12 A subset E of a metric space (S, d) is said to be open
if each point a E E has a spherical neighborhood contained in E, i.e., if
Ns(a) CE for some 8 > 0 and for every a EE. Alternatively, Eis an open
set if it is a neighborhood of each of its points.
Example In (~.2,d) the set E = {(x,y): x^2 + y^2 < r^2 } is an open set.
Definition 2.13 A subset F of a metric space (S, d) is said to be closed
iff the set F' = S - F is open. Alternatively, F is closed iff it contains all
points a with the property Ns(a) n F =f. 0 for every 8 > 0.
Example In (<C, d) the set F = {z: lzl ::::; 1} is closed.
Remark If (S, d) is a metric space, there may be subsets of S that are
neither open nor closed. For instance, in (<C, d) the set G = { z: lzl < 1} U
{ i} is neither open nor closed.
On the other hand, the empty set 0 and the whole space S = 01 are
at the same time open and closed. In certain spaces there may be subsets
(other than 0 and S) which are also open and closed. For example, if X =
{z: jzj < l}U{z: lz -3j < 1} with the metric induced on X by (<C, d), then
each of the sets {z: lzl < 1} and {z: lz -31<1}, which are complementary
in X, are both open and closed.
Theorem 2.2 The following properties hold:- The union of any collection of open subsets of a metric space is open.