Definition:Irreducible Space

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Definition

Definition 1

A topological space $T = \left({S, \tau}\right)$ is irreducible if and only if there exists no cover of $T$ by two proper closed sets of $T$.


Definition 2

A topological space $T = \left({S, \tau}\right)$ is irreducible if and only if there is no finite cover of $T$ by proper closed sets of $T$.


Definition 3

A topological space $T = \left({S, \tau}\right)$ is irreducible if and only if every two non-empty open sets of $T$ have non-empty intersection:

$\forall U, V \in \tau: U, V \ne \varnothing \implies U \cap V \ne \varnothing$


Definition 4

A topological space $T = \left({S, \tau}\right)$ is irreducible if and only if every non-empty open set of $T$ is (everywhere) dense in $T$.


Definition 5

A topological space $T = \left({S, \tau}\right)$ is irreducible if and only if the interior of every proper closed set of $T$ is empty.


Definition 6

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

$\forall U \in \tau: U^- = S$


Definition 7

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


Also known as

An irreducible space is also known as a hyperconnected space.


Also see

  • Results about irreducible spaces can be found here.