Definition:Ultraconnected Space

Definition

Definition 1

A topological space $T = \struct {S, \tau}$ is ultraconnected if and only if no two non-empty closed sets are disjoint.

Definition 2

A topological space $T = \struct {S, \tau}$ is ultraconnected if and only if the closures of every distinct pair of elements of $S$ are not disjoint:

$\forall x, y \in S: \set x^- \cap \set y^- \ne \O$

Definition 3

A topological space $T = \left({S, \tau}\right)$ is ultraconnected if and only if every closed set of $T$ is connected.

Also see

• Equivalence of Definitions of Ultraconnected Space

• Definition:Irreducible Space

• Results about ultraconnected spaces can be found here.