Long line - Revision history

Vipul: 3 revisions

2008-05-11T19:48:57Z

3 revisions

← Older revision	Revision as of 19:48, 11 May 2008
(No difference)

Vipul at 23:12, 15 December 2007

2007-12-15T23:12:02Z

← Older revision		Revision as of 23:12, 15 December 2007
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	==Definition==		==Definition==

	The '''long line''' is defined as follows: Let <math>S_\Omega</math> denote the minimal uncountable well-ordered set. Then <math>L = S_\Omega \times [0,1)</math>, in the [[dictionary order]], is the long line.		The '''long line''' is defined as follows: Let <math>S_\Omega</math> denote the [[minimal uncountable well-ordered set]]. Then <math>L = S_\Omega \times [0,1)</math>, in the [[dictionary order]], is the long line.

	==Topological space properties==		==Topological space properties==

Vipul: /* Properties it does satisfy */

2007-11-10T21:31:47Z

Properties it does satisfy

← Older revision		Revision as of 21:31, 10 November 2007
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	* [[Locally Euclidean space]]: It is in fact locally homeomorphic to <math>\R^1</math>		* [[Locally Euclidean space]]: It is in fact locally homeomorphic to <math>\R^1</math>
	* [[Normal space]]		* [[Normal space]]

			Thus the long line fails to satisfy only the second condition for a manifold; it is simply too long.

Vipul at 21:30, 10 November 2007

2007-11-10T21:30:14Z

New page

{{particular topospace}}

{{standard counterexample}}

==Definition==

The '''long line''' is defined as follows: Let <math>S_\Omega</math> denote the minimal uncountable well-ordered set. Then <math>L = S_\Omega \times [0,1)</math>, in the [[dictionary order]], is the long line.

==Topological space properties==

===Properties it does not satisfy===

* [[Second-countable space]]: The long line is ''not'' second-countable
* [[Sub-Euclidean space]]: The long line cannot be embedded inside any Euclidean space

===Properties it does satisfy===

* [[Linearly orderable space]]: It is defined using a linear order, so it is clearly linearly orderable
* [[Locally Euclidean space]]: It is in fact locally homeomorphic to <math>\R^1</math>
* [[Normal space]]