Definition:Irreducible Space/Definition 6

Definition

A topological space $T = \struct {S, \tau}$ is irreducible if and only if the closure of every non-empty open set is the whole space:

$\forall U \in \tau: U \ne \O \implies U^- = S$

Also known as

An irreducible space is also known as a hyperconnected space.

Also see

• Equivalence of Definitions of Irreducible Space

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness