Topological manifold - Revision history

Vipul: 5 revisions

2008-05-18T20:11:41Z

5 revisions

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Vipul at 20:32, 13 December 2007

2007-12-13T20:32:52Z

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			{{topospaces version at\|manifold}}

	==Definition==		==Definition==

Vipul: /* Coordinate charts and atlases */

2007-08-27T08:15:29Z

Coordinate charts and atlases

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	===Coordinate charts and atlases===		===Coordinate charts and atlases===

	A ''topological atlas''' on a topological manifold is the following data:		A '''topological atlas''' on a topological manifold is the following data:

	* An open cover of the manifold		* An open cover of the manifold

Vipul: /* Definition */

2007-06-15T04:37:53Z

Definition

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	* It is [[Hausdorff space\|Hausdorff]]		* It is [[Hausdorff space\|Hausdorff]]
	* It is [[second-~~coutanble~~ topological space\|second-~~coutanble~~]], viz it has a countable basis of open sets		* It is [[second-countable topological space\|second-countable]], viz it has a countable basis of open sets
	* Every point in the space has a neighbourhood that is homeomorphic to an open set in <math>\R^n</math>. Such a neighbourhood is termed a ''coordinate neighbourhood'' and the homoemorphism is termed a coordinate chart.		* Every point in the space has a neighbourhood that is homeomorphic to an open set in <math>\R^n</math>. Such a neighbourhood is termed a ''coordinate neighbourhood'' and the homoemorphism is termed a coordinate chart.

Vipul: /* Definition */

2007-06-15T04:37:16Z

Definition

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	A [[topological space]] is said to be a '''topological manifold''' or simply '''manifold''') of dimension <math>n</math> if it satisfies the following conditions:		A [[topological space]] is said to be a '''topological manifold''' or simply '''manifold''') of dimension <math>n</math> if it satisfies the following conditions:

	* It is [[Hausdorff space"Hausdorff]]		* It is [[Hausdorff space\|Hausdorff]]
	* It is [[second-coutanble topological space\|second-coutanble]], viz it has a countable basis of open sets		* It is [[second-coutanble topological space\|second-coutanble]], viz it has a countable basis of open sets
	* Every point in the space has a neighbourhood that is homeomorphic to an open set in <math>\R^n</math>. Such a neighbourhood is termed a ''coordinate neighbourhood'' and the homoemorphism is termed a coordinate chart.		* Every point in the space has a neighbourhood that is homeomorphic to an open set in <math>\R^n</math>. Such a neighbourhood is termed a ''coordinate neighbourhood'' and the homoemorphism is termed a coordinate chart.

Vipul at 04:36, 15 June 2007

2007-06-15T04:36:31Z

New page

==Definition==

A [[topological space]] is said to be a '''topological manifold''' or simply '''manifold''') of dimension <math>n</math> if it satisfies the following conditions:

* It is [[Hausdorff space"Hausdorff]]
* It is [[second-coutanble topological space|second-coutanble]], viz it has a countable basis of open sets
* Every point in the space has a neighbourhood that is homeomorphic to an open set in <math>\R^n</math>. Such a neighbourhood is termed a ''coordinate neighbourhood'' and the homoemorphism is termed a coordinate chart.

==Relation with other structures==

===Stronger structures===

* [[Differential manifold]]
* [[Measured manifold]]

==Facts==

===Coordinate charts and atlases===

A ''topological atlas''' on a topological manifold is the following data:

* An open cover of the manifold
* For each member of the open cover, ahomoemorphism from that open set to an open set in <math>\R^n</math>. Since this homeomorphism associates Euclidean coordinates to points in that open set, it is termed a ''coordinate chart''

===Transition function between two coordinate charts===

Suppose <math>U_1, U_2</math> are open subsets of <math>X</math>, and <math>V_1, V_2</math> open subsets of <math>\R^n</math>, with homeomorphisms <math>\phi_i: U_i \to V_i</math>. Then the transition function for these coordinate charts is defined as the map from <math>\phi_1(U_1 \cap U_2)</math> to <math>\phi_2(U_1 \cap U_2)</math> given by <math>\phi_2 \circ \phi_1^{-1}</math> (note that both these maps make sense in the area we are defining).

Intuitively, the transition function is a way to pass from one coordinate chart to another, for those points for which both coordinate charts are valid.