Manifold

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

A topological space is said to be a manifold if it satisfies the following equivalent conditions:

• It is Hausdorff
• It is second-countable
• It is locally Euclidean, viz every point has a neighbourhood that is homeomorphic to some open set in Euclidean space. If the topological space is connected, the dimension of the Euclidean space is the same for all points

Relation with other properties

Stronger properties

• Complete manifold
• Compact manifold
• Paracompact manifold
• Underlying space of a Lie group

Weaker properties

• Locally Euclidean space
• Polyhedron
• CW-space
• Perfectly normal space (and all separation axioms sitting below it)
• Locally contractible space

Metaproperties

Products

This property of topological spaces is closed under taking finite products

A direct product of manifolds is again a manifold. Fill this in later