Connected manifold

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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
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