some definitions related to open sets 3 (connected spaces)

Open sets are a most important and basic concept. We will also understand it by following explanations.

Giving a metric space $\Omega$  and subsets $A,B\subset \Omega$ .
If $A,B\ne \phi$ , $A\cap B=\phi$  and both $A$  and $B$ are open sets,
$S=A\cup B$  is not a connected space (or set).
$S$  is called a disconnected space (or set).

A connected set means we can not divide it two open sets which are not empty
and of which the intersection is empty. If $S$  is connected, $A$  or $B$  should be empty.

A trivial example of a connected space is a set which consists of a point and the empty set.
Namely, a point is the connected space.

As it is a obviously definite thing that the line of real numbers is connected,
we get to a following basic proposition.

A connected set on the real number line is a interval. (a set is not empty and the interval may be open or closed. )