# Space in which the connected components coincide with the quasicomponents

From Topospaces

## Definition

A **space in which the connected components coincide with the quasicomponents** is a topological space satisfying the following equivalent conditions:

- Each quasicomponent is connected with the subspace topology.
- Each connected component is a quasicomponent.
- Each quasicomponent is a connected component.
- The connected components and quasicomponents coincide.

## Relation with other properties

### Stronger properties

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

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

space with finitely many connected components | Space in which all connected components are open|FULL LIST, MORE INFO | |||

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

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