Ultraconnected space

In mathematics, a topological space ${\displaystyle X}$ is said to be ultraconnected if no pair of nonempty closed sets of ${\displaystyle X}$ is disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no ${\displaystyle T_{1}}$ space with more than 1 point is ultraconnected.[1]

All ultraconnected spaces are path-connected (but not necessarily arc connected[1]), normal, limit point compact, and pseudocompact.