Space in which all connected components are open
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 |