ultraconnected

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\\cap\operatorname\$, the function f: ,1to 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

See also

* Hyperconnected space

Notes

References

*
* 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).

Properties of topological spaces