Topological manifold

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

an open ball in $ℝ 𝑛$ ; or
the whole of $ℝ 𝑛$

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 $ℝ 𝑛$ 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 $𝑓 : ℝ 𝑛 + 1 \to ℝ$ with no stationary points is an $𝑛$ -dimensional manifold.

Topological manifold

Properties

See also

Footnotes