Disconnected Space

A topological space that is not connected, i.e., which can be decomposed as the disjoint union of two nonempty open subsets. Equivalently, it can be characterized as a space with more than one connected component.

A subset  of the Euclidean plane with more than one element can always be disconnected by cutting it through with a line (i.e., by taking out its intersection with a suitable straight line). In fact, it is certainly possible to find a line  such that two points of  lie on different sides of . If the Cartesian equation of  is

(1)

for fixed real numbers , then the set  is disconnected, since it is the union of the two nonempty open subsets

(2)

and

(3)

which are the sets of elements of  lying on the two sides of .