Ultraconnected space

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

Properties

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

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.

See also

• Hyperconnected space

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).