Connected manifold
This article describes a property of topological spaces obtained as a conjunction of the following two properties: connected space and manifold
This article defines a property of manifolds and hence also of topological spaces
Definition
A connected manifold is a topological space satisfying the following equiavlent conditions:
- It is a connected space that is also a manifold.
- It is a path-connected space that is also a manifold.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| compact connected manifold | connected and also a compact space | |FULL LIST, MORE INFO | ||
| simply connected manifold | manifold that is also a simply connected space | |FULL LIST, MORE INFO | ||
| compact connected orientable manifold | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| homogeneous space | connected manifold implies homogeneous | |FULL LIST, MORE INFO | ||
| manifold | |FULL LIST, MORE INFO |
See also Manifold#Weaker properties