# Ultraconnected space

In mathematics, a topological space is said to be **ultraconnected** if no pair of nonempty closed sets of is disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no 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.

## See also

## Notes

- 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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