Sorgenfrey plane - Revision history

Vipul at 17:45, 20 July 2008

2008-07-20T17:45:00Z

← Older revision		Revision as of 17:45, 20 July 2008
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	* [[Separable space]]: This is because the Sorgenfrey line is separable, and a finite product of separable spaces is again separable.		* [[Separable space]]: This is because the Sorgenfrey line is separable, and a finite product of separable spaces is again separable.

			==References==
			===Textbook references===
			* {{booklink-defined\|Munkres}}, Page 198, Example 3, Chapter 4, Section 31

Vipul: 2 revisions

2008-05-11T19:58:55Z

2 revisions

← Older revision	Revision as of 19:58, 11 May 2008
(No difference)

Vipul: /* Properties it does not satisfy */

2007-12-15T22:40:57Z

Properties it does not satisfy

← Older revision		Revision as of 22:40, 15 December 2007
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	* [[Lindelof space]]: The Sorgenfrey plane is ''not'' Lindelof, even though the Sorgenfrey line is Lindelof.		* [[Lindelof space]]: The Sorgenfrey plane is ''not'' Lindelof, even though the Sorgenfrey line is Lindelof.
	* [[Normal space]]: The Sogenfrey plane is ''not'' normal, even though the Sorgenfrey line is normal.		* [[Normal space]]: The Sogenfrey plane is ''not'' normal, even though the Sorgenfrey line is normal. {{proofat\|[[Sorgenfrey plane is not normal]]}}
	* [[Hereditarily separable space]]: The anitdiagonal in the Sorgenfrey plane is a discrete uncountable set.		* [[Hereditarily separable space]]: The anitdiagonal in the Sorgenfrey plane is a discrete uncountable set.

Vipul at 19:41, 15 December 2007

2007-12-15T19:41:40Z

New page

{{standard counterexample}}

==Definition==

The Sorgenfrey plane is defined as the Cartesian product of two copies of the [[Sorgenfrey line]], endowed with the [[product topology]].

==Topological space properties==

===Properties it does not satisfy===

* [[Lindelof space]]: The Sorgenfrey plane is ''not'' Lindelof, even though the Sorgenfrey line is Lindelof.
* [[Normal space]]: The Sogenfrey plane is ''not'' normal, even though the Sorgenfrey line is normal.
* [[Hereditarily separable space]]: The anitdiagonal in the Sorgenfrey plane is a discrete uncountable set.

===Properties it does satisfy===

* [[Separable space]]: This is because the Sorgenfrey line is separable, and a finite product of separable spaces is again separable.