104 Chapter^2the polygonal line, call it I, must join a point in G 1 to a point in G2. Now
and since G 1 and G 2 are separated, so are G 1 n I and G2 n J. But this is
impossible, since I is connected. Hence A is connected.
Remarks Theorem 2.11 generalizes to :!Rn (n > 2).
The theorem does not hold for closed sets. For instance, the circle C ={z: lzl = 1} is connected, yet no two points can be joined by a polygonal
line lying in C.
Definition 2.32 A nonempty connected open set is called a region or a
domain. The closure of a region is called a closed region. Sometimes a
region, as defined above, is more specifically referred to as an open region,
the term region being applied more generally to any open (nonempty)
connected set together with some, none, or all of its boundary points.
Examples The whole plane, a half-plane, the interior of a triangle, and
an open disk are all regions.
Theorem 2.12 In :!Rn the components of any open set are open, and the
collection of its components is countable.
Proof Let A be an open set in :!Rn, and let }( be any component of A. To
prove that }(is open, consider an arbitrary point p EK. Then p EA and
since A is open, there is a neighborhood N 0 (p) C A. For any, q E N 0 (p) we
can join p and q by a line segment contained in A. Hence N 0 (p) belongs
to the component of A that contains p, so that N 0 (p) C K. This shows
that }( is open.
Clearly, every open set in :!Rn contains a point (x 1 ,x 2 , ••• ,xn) with ra-
tional coordinates. It is known that the set of points in :!Rn with rational
coordinates is countable, so its points can be arranged into a sequence p.j }.
For each component }( of A there is a smallest j such that Aj E K. Since
the components of A are disjoint sets, to different components correspond
different values of j. Hence the set of the components of A can be put
into one-to-one correspondence with a subset of the natural numbers, so
it is a countable set.
Corollary 2.1 Any open set on JR^1 is the union of a countable collection
of disjoint open intervals.
Remark The components of an open set in a general metric space need
not be open.