# Connected manifold

From Topospaces

*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*

## Contents

## 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 | |||

homology manifold | locally compact space whose homology groups with respect to the exclusion of any point look like those of a manifold | Manifold|FULL LIST, MORE INFO | ||

manifold with boundary | Hausdorff, second-countable, and every point is contained in an open subset that is homeomorphic to an open subset of Euclidean half-space | Manifold|FULL LIST, MORE INFO | ||

closed sub-Euclidean space | homeomorphic to a closed subset of Euclidean space | manifold implies closed sub-Euclidean | obvious counterexamples, such as a closed unit disk | |FULL LIST, MORE INFO |

metrizable space | underlying topological space of a metric space | Manifold|FULL LIST, MORE INFO | ||

paracompact Hausdorff space | paracompact and Hausdorff | (via metrizable) | Manifold|FULL LIST, MORE INFO | |

normal space | any two disjoint closed subsets can be separated by disjoint open subsets | Manifold|FULL LIST, MORE INFO | ||

regular space | any point and closed subset not containing it can be separated by disjoint open subsets | Manifold|FULL LIST, MORE INFO | ||

Hausdorff space | any two distinct points can be separated by disjoint open subsets | Manifold|FULL LIST, MORE INFO | ||

locally Euclidean space | every point is contained in an open subset that is homeomorphic to an open subset of Euclidean space | Manifold|FULL LIST, MORE INFO | ||

locally contractible space | it has a basis of open subsets that are all contractible | Manifold|FULL LIST, MORE INFO | ||

locally metrizable space | it has a basis of open subsets that are all metrizable | Manifold|FULL LIST, MORE INFO | ||

locally compact space | every point is contained in an open subset whose closure is compact | Manifold|FULL LIST, MORE INFO | ||

nondegenerate space | the inclusion of any point in it is a cofibration | manifold implies nondegenerate | Manifold|FULL LIST, MORE INFO | |

compactly nondegenerate space | every point is contained in an open subset whose closure is compact, and the inclusion of the point in the closure is a cofibration. | Manifold|FULL LIST, MORE INFO |

See also Manifold#Weaker properties