Ultraconnected space

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 ${\displaystyle X}$ is path-connected (but not necessarily arc connected). If ${\displaystyle a}$ and ${\displaystyle b}$ are two points of ${\displaystyle X}$ and ${\displaystyle p}$ is a point in the intersection ${\displaystyle \operatorname {cl} \{a\}\cap \operatorname {cl} \{b\}}$, the function ${\displaystyle f:[0,1]\to X}$ defined by ${\displaystyle f(t)=a}$ if ${\displaystyle 0\leq t<1/2}$, ${\displaystyle f(1/2)=p}$ and ${\displaystyle f(t)=b}$ if ${\displaystyle 1/2<t\leq 1}$, is a continuous path between ${\displaystyle a}$ and ${\displaystyle b}$.^[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

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

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).