Manifold

Topological manifold

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

• an open ball in ℝ𝑛; or
• the whole of ℝ𝑛

Thus the so-called Euclidean balls form a topological basis of the entire manifold 𝑀. A homeomorphism between a Euclidean neighbourhood and an open subset of ℝ𝑛 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 𝑓 :ℝ𝑛+1 →ℝ with no stationary points is an 𝑛-dimensional manifold.

See also

• [[Category of manifolds|𝖬𝖺𝗇0]]

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Footnotes

1. German topologische Mannigfaltigkeit. ↩