In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint.PlanetMath Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T1 space with more than one point is ultraconnected.Steen & Seebach, Sect. 4, pp. 29-30
Properties
Every ultraconnected space is path-connected (but not necessarily arc connected). If and are two points of and is a point in the intersection , the function defined by if , and if , is a continuous path between and .
Every ultraconnected space is normal, limit point compact, and pseudocompact.
Examples
The following are examples of ultraconnected topological spaces. A set with the indiscrete topology. The Sierpiński space. A set with the excluded point topology. The right order topology on the real line.Steen & Seebach, example #50, p. 74
See also Hyperconnected space
Notes
References
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).