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 S 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 r such that two points of S lie on different sides of r. If the Cartesian equation of r is


for fixed real numbers a,b,c, then the set S^'=S\r is disconnected, since it is the union of the two nonempty open subsets

 U_+=S^' intersection {(x,y) in R^2|ax+by+c>0}


 U_-=S^' intersection {(x,y) in R^2|ax+by+c<0},

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

See also

Totally Disconnected Space

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Disconnected Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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