First-countable space - Revision history

Vipul: 10 revisions

2008-05-11T19:44:46Z

10 revisions

← Older revision	Revision as of 19:44, 11 May 2008
(No difference)

2008-04-21T22:23:19Z

Textbook references

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	===Textbook references===		===Textbook references===
	* {{booklink\|Munkres}}, Page 190 (formal definition)		* {{booklink\|Munkres}}, Page 190 (formal definition)
			* {{booklink\|SingerThorpe}}, Page 39 (formal definition)

2008-04-21T22:20:45Z

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	If every point has a neighbourhood which is first-countable, then the whole topological space is first-countable.		If every point has a neighbourhood which is first-countable, then the whole topological space is first-countable.

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 190 (formal definition)

2008-02-02T01:16:21Z

Weaker properties

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	===Weaker properties===		===Weaker properties===

			* [[Compactly generated space]]: {{proofat\|[[First-countable implies compactly generated]]}}
	* [[Sequential space]]		* [[Sequential space]]

2007-12-17T19:42:29Z

Stronger properties

2007-12-17T19:36:18Z

Metaproperties

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	Any countable product of first-countable spaces is first-countable.		Any countable product of first-countable spaces is first-countable.

			{{local}}

			If every point has a neighbourhood which is first-countable, then the whole topological space is first-countable.

2007-12-17T19:35:09Z

Definition with symbols

← Older revision		Revision as of 19:35, 17 December 2007
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	===Definition with symbols===		===Definition with symbols===

	A [[topological space]] <math>X</math> is said to be '''first-countable''' if for any <math>x \in X</math>, there exists a countable collection ,math>U_n</math> of open sets around <math>x</math> such that any open <math>V \ni x</math> contains some <math>U_n</math>.		A [[topological space]] <math>X</math> is said to be '''first-countable''' if for any <math>x \in X</math>, there exists a countable collection <math>U_n</math> of open sets around <math>x</math> such that any open <math>V \ni x</math> contains some <math>U_n</math>.

	==Relation with other properties==		==Relation with other properties==

2007-11-10T23:55:09Z

2007-11-10T23:41:02Z

2007-11-10T23:38:26Z

← Older revision		Revision as of 23:38, 10 November 2007
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	* [[Sequential space]]		* [[Sequential space]]

			==Metaproperties==

			{{subspace-closed}}

			Any subspace of a first-countable space is first-countable. We can take, for our new basis at any point, the intersection of the old basis elements with the subspace. {{proofat\|[[First-countability is hereditary]]}}

			{{countable DP-closed}}

			Any countable product of first-countable spaces is first-countable.