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 $$X$$ 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.