Definition:Irreducible Space

Definition

Definition 1

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

Definition 2

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

Definition 3

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

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

Definition 4

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

Definition 5

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

Definition 6

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:

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

Definition 7

A topological space $T = \struct {S, \tau}$ 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

• Equivalence of Definitions of Irreducible Space
• Definition:Irreducible Component
• Definition:Ultraconnected Space

• Results about irreducible spaces can be found here.

Linguistic Note

The thinking behind applying the word irreducible to the concept of an irreducible space arises as follows.

By definition, we cannot express $X$ as the union of two proper closed sets of $T$.

Expressing a space as the union of two smaller closed sets can be considered as reducing it.

There are parallels with the concept of an irreducible element of a ring in abstract algebra, which cannot be written as a product of two non-units.

The terminology comes from the Zariski topology in the context of algebraic geometry, where there is a direct link to irreducible varieties.

While the name hyperconnected space is more intuitively clear, and bears a pleasing antithesis with the concept of ultraconnected space, it is considered old-fashioned.