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2008-05-11T19:43:36Z

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Vipul at 16:18, 2 December 2007

2007-12-02T16:18:49Z

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{{topospace point property}}

==Definition==

A point in a [[topological space]] is termed a <math>m</math>-'''Euclidean point''' if it satisfies the following equivalent conditions:

* It has a neighbourhood (open set containing it) that is homeomorphic to <math>\R^m</math>
* It has a neighbourhood (open set containing it) that is homeomorphic to an open set in <math>\R^m</math>
* Every neighbourhood of it contains a smaller neighbourhood homeomorphic to an open set in <math>\R^m</math>
* Every neighbourhood of it contains a smaller neighbourhood homeomorphic to <math>\R^m</math>

A point is Euclidean if it is <math>m</math>-Euclidean for some <math>m</math>. A point cannot be <math>m</math>-Euclidean and <math>n</math>-Euclidean for <math>m \ne n</math>.
==Relation with other properties==

===Stronger properties===

* [[Hausdorff-Euclidean point]]
* [[Closed Euclidean point]]

==Facts==

* In a [[locally Euclidean space]], and more specifically in a [[manifold]], every point is Euclidean (in fact, since locally Euclidean spaces are [[T1 space|T1]], every point is closed Euclidean).

Euclidean point - Revision history

Vipul: 1 revision

Vipul at 16:18, 2 December 2007