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2014/04/26

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. )




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