# Space in which all connected components are open

From Topospaces

## Contents

## Definition

A **space in which all connected components are open** is a topological space satisfying the following equivalent conditions:

- Every point is contained in an open subset that is connected in the subspace topology.
- Every connected component is an open subset.
- Every connected component is a clopen subset.

## Relation with other properties

### Stronger properties

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

connected space | ||||

locally connected space | ||||

space with finitely many connected components |

### Weaker properties

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

space in which the connected components coincide with the quasicomponents |