Manifold

Topological manifold

An -dimensional topological manifold¹ is a second-countable Hausdorff space that locally resembles -dimensional Real coördinate space , i.e. every has an open neighbourhood that is homeomorphic to a open subset of . topology These neighbourhoods are called Euclidean neighbourhoods of the manifold. Without loss of generality, every point has a neighbourhood homeomorphic to either

• an open ball in ; or
• the whole of

Thus the so-called Euclidean balls form a topological basis of the entire manifold . A homeomorphism between a Euclidean neighbourhood and an open subset of is called a chart, and a set of charts covering the whole manifold is called an atlas. A Transition map allows for the transition between overlapping charts. Topological manifolds are the most basic kind of Manifold; every manifold is topologically a manifold.

Properties

• Every manifold is a Locally compact space.
• A Level set of a multivariable function with no stationary points is an -dimensional manifold.

See also

• [[Category of manifolds|]]

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Footnotes

1. German topologische Mannigfaltigkeit. ↩