In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint.
[1] 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.
[2] Every ultraconnected space
is path-connected (but not necessarily arc connected).
is a point in the intersection
cl { a } ∩ cl { b }
, is a continuous path between
[2] Every ultraconnected space is normal, limit point compact, and pseudocompact.
[1] The following are examples of ultraconnected topological spaces.