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Path: blob/main/01Manifold_Def.ipynb
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1. Basic notions of topology
This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
A topology on a set is a collection of subsets containing both the empty set ∅ and the set such that is closed under arbitrary unions and finite intersections, i.e.,
(i) if for all in an index set , then ,
(ii) if , then
The elements of are called open sets. The set with a topology will be called a topological space.
If is a subset of a topological space , then the subspace topology on is defined as
A neighborhood of a point in is an open set containing .
A subcollection of a topology on a topological space is a basis for the topology if given an open set and point , there is an open set such that . We also say that generates the topology or that is a basis for the topological space . A collection of open sets of is a basis if and only if every open set in is a union of sets in .
If are topological spaces a function is continuous if and only if the inverse image of any open set is open.
A continuous bijection whose inverse is also continuous is called a homeomorphism.
Topologigal manifold
A topological manifold of dimension is a topological space with the following properties:
(i) is Hausdorff, that is, for each pair of distinct points of there exist neighborhoods of and , respectively such that .
(ii) Each point possesses a neighborhood homeomorphic to an open subset of .
(iii) satisfies the second countability axiom, that is, has a countable basis for its topology.
For an -dimensional topological manifold each pair , where is an open subset of and is a homeomorphism of to an open subset of is called a coordinate map, chart or coordinate system and is a coordinate neighborhood.
For belongs to , and therefore consists of real numbers that depend on . Thus is of the form
Smooth manifold
A map from an open subset to is smooth on or belongs to if all partial derivatives are continuous on .
When two coordinate neighborhoods overlap, we have formulas for the associated coordinate change. The idea to obtain smooth manifolds is to choose a subcollection of coordinate neighborhoods so that the coordinate changes are smooth maps.
An -dimensional or smooth manifold is a topological manifold of dimension and a family of coordinate charts defined on open sets , such that:
(i) the coordinate neighborhoods cover ,
(ii) for each pair of indices such that the overlap maps (transitions)
are ,
(iii) the family is maximal with respect to (i) and (ii), meaning that if is a chart such that and are for all , then is in .
Any family that satisfies (i) and (ii) is called a -atlas for . If also satisfies (iii) it is called a maximal atlas or a differentiable or smooth structure.
Given any atlas on , there is a unique maximal atlas containing it. In fact, we can take the set of all maps that satisfy (ii) with every coordinate neighborhoods on . Clearly , and one can easily check that satisfies (i) and (ii). Also, by construction, is maximal with respect to (i) and (ii). Two atlases are said to be equivalent if they define the same differentiable structure.
Smooth functions and maps.
By we shall denote the family of smooth functions on a smooth manifold , i.e., functions , such that is smooth on for every coordinate chart .
If , are smooth manifolds, then a map is smooth if for every pair of charts on and on , the map is smooth on
What's next?
Take a look at the notebook Examples of charts. Cartesian and spherical coordinates.