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Ultraconnected space

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In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint.[1] 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.[2]

Properties

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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 .[2]

Every ultraconnected space is normal, limit point compact, and pseudocompact.[1]

Examples

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The following are examples of ultraconnected topological spaces.

See also

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Notes

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  1. ^ a b PlanetMath
  2. ^ a b Steen & Seebach, Sect. 4, pp. 29-30
  3. ^ Steen & Seebach, example #50, p. 74

References

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  • 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).