Topology of Plane Sets of Points 101
To see that the condition is not necessary, consider in C the sets A =
{z: lzl < 1} and B = {z: lz -21<1}, which are separated, yet at a zero
distance.
(6) Since A is closed, we have A = A, orBUC=BUC=BUC
and it follows that
B = B n (Bu C) = B n (Bu C) = (B n B) u (B n C) = B
so that B is closed. Similarly, C is closed.Definition 2.30 Let (S, d) be a metric space and A C S. A is said to
be disconnected if it is the union of two nonempty separated sets. A is
called connected if it is not disconnected. Thus a subset of a metric space
is connected if it. cannot be represented as the union of two nonempty
separated sets.
By using Theorem 2.7 it follows that the space (S, d) itself is connected
if it cannot be represented as the union of two nonempty disjoint open sets,
or as the union of two nonempty disjoint closed sets. This characterization
may also be applied to any subset A of S by considering A as a metric
space with the relative metric. Then it follows that A is connected iff the
only nonempty subset of A that is both open and closed with respect to
the subspace A is A itself. Also, A C S is disconnected iff A C G 1 U G 2 ,
where G1 and G 2 are disjoint open sets, or disjoint closed sets, such that
A n G 1 and A n G 2 are nonempty.
The empty set, as well as any singleton, are trivially connected sets.
Theorem 2.8 Connected sets have the following properties:
1. If A is connected, then A is connected.
- if A is connected and B is such that A C B C A, then B is connected.
3. If A and B are connected and if they are not separated, then A U B
is connected.- If A and B are connected and if An B # 0, then AU Bis connected.
More generally, if {A,,} a EI is a family of connected sets such that
naEI Aa # 0, then UaEI Aa is connected.
5. If every pair of points of A lies in a connected subset of A, then A
·is connected.
Proof We shall prove properties 1 and 5, leaving the proofs of the other
parts to the interested reader.
(1) Suppose that A is disconnected. Then we have A = F1 U F2, where
F 1 and F 2 are disjoint nonempty closed sets. This implies that