Ultraconnected space

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, a topological space X is said to be ultraconnected if no pair of nonempty closed sets of X is disjoint. Equivalently, a space is ultraconnected if and only if the closures of two disjoint points always have non trivial intersection. Hence, no 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.

Notes[edit]

  1. ^ a b Steen and Seeback, Sect. 4

See also[edit]

References[edit]