Difference between revisions of "Topological manifold"
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| − | '''Note:''' This page refers to a '''Topological Manifold''' | + | '''Note:''' This page refers to a '''Topological Manifold''' a special kind of [[Manifold]] |
==Definition== | ==Definition== | ||
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# Let {{M|M}} be an {{M|n-}}manifold | # Let {{M|M}} be an {{M|n-}}manifold | ||
# Let {{M|M^n}} be a manifold | # Let {{M|M^n}} be a manifold | ||
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| + | ==See also== | ||
| + | * [[Chart]] | ||
| + | * [[Atlas]] | ||
| + | * [[Manifolds]] | ||
==References== | ==References== | ||
Latest revision as of 01:13, 6 April 2015
Note: This page refers to a Topological Manifold a special kind of Manifold
Contents
Definition
We say [ilmath]M[/ilmath] is a topological manifold of dimension [ilmath]n[/ilmath] or simply an [ilmath]n-[/ilmath]manifold if it has the following properties[1]:
- [ilmath]M[/ilmath] is a Hausdorff space - that is for every pair of distinct points [ilmath]p,q\in M\ \exists\ U,V\subseteq M\text{ (that are open) } [/ilmath] such that [ilmath]U\cap V=\emptyset[/ilmath] and [ilmath]p\in U,\ q\in V[/ilmath]
- [ilmath]M[/ilmath] is Second countable - there exists a countable basis for the topology of [ilmath]M[/ilmath]
- [ilmath]M[/ilmath] is locally Euclidean of dimension [ilmath]n[/ilmath] - each point of [ilmath]M[/ilmath] has a neighbourhood that his homeomorphic to an open subset of [ilmath]\mathbb{R}^n[/ilmath]
- This actually means that for each [ilmath]p\in M[/ilmath] we can find:
- an open subset [ilmath]U\subseteq M[/ilmath] with [ilmath]p\in U[/ilmath]
- an open subset [ilmath]\hat{U}\subseteq\mathbb{R}^n[/ilmath]
- and a Homeomorphism [ilmath]\varphi:U\rightarrow\hat{U} [/ilmath]
- This actually means that for each [ilmath]p\in M[/ilmath] we can find:
Notations
The following are all equivalent (most common first):
- Let [ilmath]M[/ilmath] be a manifold of dimension [ilmath]n[/ilmath]
- Let [ilmath]M[/ilmath] be an [ilmath]n-[/ilmath]manifold
- Let [ilmath]M^n[/ilmath] be a manifold
See also
References
- ↑ John M Lee - Introduction to smooth manifolds - Second Edition
