Ultraconnected space

In mathematics, a topological space $X$ is said to be ultraconnected if no pair of nonempty closed sets of $X$ is disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no $T_1$ space with more than 1 point is ultraconnected.^[1]

All ultraconnected spaces are path-connected (but not necessarily arc connected^[1]), normal, limit point compact, and pseudocompact.

Notes

↑ 1.0 1.1 Steen and Seeback, Sect. 4

See also

Hyperconnected space

References

This article incorporates material from Ultraconnected space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).