# Definition:Irreducible Space

## Contents

## 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

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

- Results about
**irreducible spaces**can be found here.