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 is path-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.
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.