Ultraconnected space

In mathematics, a topological space ${\displaystyle X}$ is said to be ultraconnected if no pair of nonempty closed sets of ${\displaystyle X}$ is disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no ${\displaystyle T_{1}}$ 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. Steen and Seeback, Sect. 4

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).
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