# 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:

1. It is a connected space that is also a manifold.
2. 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

### Weaker properties

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