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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

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

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.^[3]

See also

• Hyperconnected space

Notes

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).