Topology of Plane Sets of Points 105
E~ample The set Q of the rationals in IB.^1 with the relative metric is open
and totally disconnected: i.e., its components are the points of Q, each of
which is a closed set.Definition 2.33 A metric space (S, d) is locally connected if every neigh-
borhood of each point p E S contains a connected neighborhood of p.
The property of local connectedness neither implies nor is implied by
connectedness in the large (i.e., as in Definition 2.30).
Theorem 2.13 Every open set in a locally connected separable space is
a countable' union of disjoint regions.
EXERCISES 2.5
1. If A and B are open sets, prove that A and B are separated iff they
are disjoint. Similarly, if A and B are closed sets, then A and B are
separated iff they are disjoint.2. If A is open and A= BU C, where B and Care separated, prove that
B and C are open.3. If A and B are closed sets, prove that An B' and B n A' are separated.
4. Let A and B be two closed sets in a metric space, If AU B and A n B
are connected sets, show that A is connected.- Prove that the union of any collection of connected sets with a common
point is connected. - Show that a 5-neighborhood in IB.n is connected.
- Let p and q be any two points of a subset A of a metric space. Define
p "' q iff there is a connected subset of A containing both p and q.
Prove that this is an equivalence relation. Show that the collection of
the equivalence classes defined by this relation gives the partition of
the set A into its components.
*8. Prove that in Theorem 2.11 it is possible to make use of polygonal lines
with sides pi;i,rallel to the coordinate axes. - Show that any convex set is connected.
- Show that any starlike open set in C is a region.
- Suppose that A is a starlike region. Prove that the set of star centers
of A is a convex closed subset of A. - Any nonempty subs~t M of C is contained in a convex set, namely,
C itself. The convex hull (or convex cover) of M is defined to be the
intersection of all convex sets that contain M. Show that the convex
hull of M is the smallest convex set that contains M. - Show that the intersection of two convex regions is convex, but the
union of two convex regions need not be convex.