[[Manifold]]
# Topological manifold

An $n$-dimensional **topological manifold**[^de] is a [[Second countability axiom|second-countable]] [[Hausdorff space]] $M$ that locally resembles $n$-dimensional [[Real coördinate space]] $\mathbb{R}^n$,
i.e. every $x \in M$ has an open [[neighbourhood]] that is [[Homeomorphism|homeomorphic]] to a open subset of $\mathbb{R}^n$. #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 $\mathbb{R}^n$; or
- the whole of $\mathbb{R}^n$

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 $\mathbb{R}^n$ 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.

[^de]: German _topologische Mannigfaltigkeit_.

## Properties

- Every manifold is a [[Locally compact space]].
- A [[Level set]] of a multivariable function $f : \mathbb{R}^{n+1} \to \mathbb{R}$ with no stationary points is an $n$-dimensional manifold.

## See also

- [[Category of manifolds|$\cat{Man}^0$]]

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