The Real And Complex Number Systems

a in S and every r 0, the set x : dx, a r is nonempty.
Proof: Assume that x : dx, a r is empty. Denote two sets x : dx, a r by A
and x : dx, a r by B. Then we have
1. A since a A and B since S is unbounded,
2. A B ,
3. A B S,
4. A Ba; r is open in S,
and consider B as follows. Since x : dx, a r is closed in S, B S x : dx, a r
is open in S. So, we know that S is disconnected which is absurb. Hence, we know that the
set x : dx, a r is nonempty.
Supplement on a connected metric space
Definition Two subsets A and B of a metric space X are said to be separated if both
A clB and clA B .
AsetE X is said to be connected if E is not a union of two nonempty separated
sets.
We now prove the definition of connected metric space is equivalent to this definiton
as follows.
Theorem A set E in a metric space X is connected if, and only if E is not the union of
two nonempty disjoint subsets, each of which is open in E.
Proof: () Suppose that E is the union of two nonempty disjoint subsets, each
of which is open in E, denote two sets, U and V. Claim that
U clV clU V .
Suppose NOT, i.e., x U clV. Thatis,thereisa 0 such that
B X x, E B E x, U and B X x, V
which implies that
B X x, V B X x, V E
B X x, E V
U V ,
a contradiction. So, we have U clV . Similarly for clU V . So,X is
disconnected. That is, we have shown that if a set E in a metric space X is
connected, then E is not the union of two nonempty disjoint subsets, each of which
is open in E.
() Suppose that E is disconnected, then E is a union of two nonempty
separated sets, denoted E A B, whereA clB clA B . Claim that A
and B are open in E. Suppose NOT, it means that there is a point x A which is
not an interior point of A. So, for any ball B E x, r, there is a correspounding
x r B, wherex r B E x, r. It implies that x clB which is absurb with
A clB . So, we proved that A is open in E. Similarly, B is open in E. Hence,
we have proved that if E is not the union of two nonempty disjoint subsets, each of
whichisopeninE, then E in a metric space X is connected.
Exercise Let A and B be connected sets in a metric space with A B not connected and
suppose A B C 1 C 2 where clC 1 C 2 C 1 clC 2 . Show that
B C 1 is connected.