Space with finitely many connected components
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 |