# Space with finitely many connected components

From Topospaces

## Contents

## Definition

A **space with finitely many connected components** is a topological space satisfying the following equivalent conditions:

- It has finitely many connected components.
- It can be expressed as a disjoint union of finitely many pairwise disjoint clopen subsets each of which is a connected space with the subspace topology.
- It has finitely many clopen subsets (the clopen subsets will precisely be all possible unions of the connected components).

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

connected space | |FULL LIST, MORE INFO | |||

space with finitely many irreducible components | |FULL LIST, MORE INFO | |||

irreducible space | Connected space|FULL LIST, MORE INFO | |||

Noetherian space | |FULL LIST, MORE INFO | |||

space with finitely many path components | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

space with finitely many quasicomponents | |FULL LIST, MORE INFO | |||

space in which all connected components are open | |FULL LIST, MORE INFO | |||

space in which the connected components concide with the quasicomponents | Space in which all connected components are open|FULL LIST, MORE INFO |