Ultraconnected space

In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint.PlanetMath 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.Steen & Seebach, Sect. 4, pp. 29-30

Properties

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

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

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.Steen & Seebach, example #50, p. 74

Notes

References

{{PlanetMath attribution|id=5814|title=Ultraconnected space}}
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).

Category:Properties of topological spaces

Ultraconnected space

Properties

Examples

See also

Notes

References