# 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:

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

### 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.