Editing Atlas (topology) (section)

==Formal definition of atlas==
An '''atlas''' for a [[topological space]] <math>M</math> is an [[indexed family]] <math>\{(U_{\alpha}, \varphi_{\alpha}) : \alpha \in I\}</math> of charts on <math>M</math> which [[Cover (topology)|covers]] <math>M</math> (that is, <math display="inline">\bigcup_{\alpha\in I} U_{\alpha} = M</math>). If for some fixed ''n'', the [[image (mathematics)|image]] of each chart is an open subset of ''n''-dimensional [[Euclidean space]], then <math>M</math> is said to be an ''n''-dimensional [[manifold]]. 

The plural of atlas is ''atlases'', although some authors use ''atlantes''.<ref>{{cite book|url=https://books.google.com/books?id=VRz2CAAAQBAJ&pg=PA1| title=Riemannian Geometry and Geometric Analysis|first=Jürgen|last=Jost|date=11 November 2013| publisher=Springer Science & Business Media|isbn=9783662223857|access-date=16 April 2018|via=Google Books}}</ref><ref>{{cite book| url=https://books.google.com/books?id=_ZT_CAAAQBAJ&pg=PA418|title=Calculus of Variations II|first1=Mariano|last1=Giaquinta| first2=Stefan|last2=Hildebrandt|date=9 March 2013|publisher=Springer Science & Business Media|isbn=9783662062012|access-date=16 April 2018|via=Google Books}}</ref>

An atlas <math>\left( U_i, \varphi_i \right)_{i \in I}</math> on an <math>n</math>-dimensional manifold <math>M</math> is called an '''adequate atlas''' if the following conditions hold:{{clarify|reason=why not restricting the charts to subsets whose images are unit balls, that is, defining adequate as "locally finite cover by open charts whose images are unit open balls"|date=May 2024}}

* The [[image (mathematics)|image]] of each chart is either <math>\R^n</math> or <math>\R_+^n</math>, where <math>\R_+^n</math> is the [[closed half-space]],{{clarify|reason=the image of a chart must be open|date=May 2024}}
* <math>\left( U_i \right)_{i \in I}</math> is a [[Locally finite collection|locally finite]] open cover of <math>M</math>, and
* <math display="inline">M = \bigcup_{i \in I} \varphi_i^{-1}\left( B_1 \right)</math>, where <math>B_1</math> is the open ball of radius 1 centered at the origin.

Every [[second-countable]] manifold admits an adequate atlas.<ref name="Kosinski 2007">{{cite book | last=Kosinski | first=Antoni | title=Differential manifolds | publisher=Dover Publications | location=Mineola, N.Y | year=2007 | isbn=978-0-486-46244-8 | oclc=853621933 }}</ref> Moreover, if <math>\mathcal{V} = \left( V_j \right)_{j \in J}</math> is an open covering of the second-countable manifold <math>M</math>, then there is an adequate atlas <math>\left( U_i, \varphi_i \right)_{i \in I}</math> on <math>M</math>, such that <math>\left( U_i\right)_{i \in I}</math> is a [[Refinement of a cover|refinement]] of <math>\mathcal{V}</math>.<ref name="Kosinski 2007" />