Connected regular space - Revision history

Vipul: 3 revisions

2008-05-11T19:42:08Z

3 revisions

← Older revision	Revision as of 19:42, 11 May 2008
(No difference)

Vipul at 17:18, 22 January 2008

2008-01-22T17:18:51Z

← Older revision		Revision as of 17:18, 22 January 2008
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	==Facts==		==Facts==

	* Any connected regular space which has at least two points, is uncountable. The proof of this is roundabout: we first observe that any [[countable space]] is [[Lindelof space\|Lindelof]], then the fact that any regular Lindelof space is normal, and then the fact that any [[connected normal space]] with at least two points has cardinality at least that of the reals. Note that this does ''not'' prove that any connected regular space has cardinality at least that of the reals.		* Any connected regular space which has at least two points, is uncountable. The proof of this is roundabout: we first observe that any [[countable space]] is [[Lindelof space\|Lindelof]], then the fact that any regular Lindelof space is normal, and then the fact that any [[connected normal space]] with at least two points has cardinality at least that of the reals. Note that this does ''not'' prove that any connected regular space has cardinality at least that of the reals. {{further\|[[connected regular implies uncountable]]}}

Vipul at 22:07, 15 December 2007

2007-12-15T22:07:31Z

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	{{topospace property conjunction\|regularity\|connectedness}}		{{topospace property conjunction\|regularity\|connectedness}}

			{{variationof\|connectedness}}

	==Definition==		==Definition==

Vipul at 22:07, 15 December 2007

2007-12-15T22:07:14Z

New page

{{topospace property conjunction|regularity|connectedness}}

==Definition==

A '''connected regular space''' is a [[topological space]] which is both [[connected space|connected]] and [[regular space|regular]].

==Facts==

* Any connected regular space which has at least two points, is uncountable. The proof of this is roundabout: we first observe that any [[countable space]] is [[Lindelof space|Lindelof]], then the fact that any regular Lindelof space is normal, and then the fact that any [[connected normal space]] with at least two points has cardinality at least that of the reals. Note that this does ''not'' prove that any connected regular space has cardinality at least that of the reals.