Definition:Ultraconnected Space

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Definition

Definition 1

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


Definition 2

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

$\forall x, y \in S: \left\{{x}\right\}^- \cap \left\{{y}\right\}^- \ne \varnothing$


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

  • Results about ultraconnected spaces can be found here.


Sources