Normal Hausdorff space - Revision history

Vipul at 22:01, 27 January 2012

2012-01-27T22:01:34Z

← Older revision		Revision as of 22:01, 27 January 2012
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! No. !! Shorthand !! A topological space is said to be normal if ... !! A topological space <math>X</math> is said to be normal if ...		! No. !! Shorthand !! A topological space is said to be normal Hausdorff if ... !! A topological space <math>X</math> is said to be normal Hausdorff if ...
	\|-		\|-
	\| 1 \|\| separation of disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given any two disjoint [[closed subset]]s in the topological space, there are disjoint open sets containing them. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any two closed subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, B \subseteq V</math>, and <math>U \cap V = \varnothing</math>.		\| 1 \|\| separation of disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given any two disjoint [[closed subset]]s in the topological space, there are disjoint open sets containing them. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any two closed subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, B \subseteq V</math>, and <math>U \cap V = \varnothing</math>.
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	Note that in each of the definitions, the [[T1 space]] assumption (that points are closed) can be replaced by the (a priori stronger) [[Hausdorff space]] assumption, without changing the meaning of the overall definition.		Note that in each of the definitions, the [[T1 space]] assumption (that points are closed) can be replaced by the (a priori stronger) [[Hausdorff space]] assumption, without changing the meaning of the overall definition.

	Some ~~versions of~~ the ~~definition omit the condition that points are closed subsets (in the above definition, the initial clause~~ ''~~all points are closed'' would be removed from ''each~~'' ~~of the above definitions). This~~ is a ~~''different''~~, ~~and~~ ''~~weaker~~''~~, notion, and is covered here as~~ [[~~normal-minus-Hausdorff~~ space]].		Some people use the term ''normal space'' for what is called here a normal Hausdorff space; however, we define the term [[normal space]] as ''not'' having the [[T1 space]] assumption.

	===Equivalence of definitions===		===Equivalence of definitions===

Vipul: moved Normal space to Normal Hausdorff space

2012-01-27T18:03:01Z

moved Normal space to Normal Hausdorff space

← Older revision	Revision as of 18:03, 27 January 2012
(No difference)

Vipul: /* Definition */

2012-01-24T22:30:26Z

Definition

← Older revision		Revision as of 22:30, 24 January 2012
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	\| 4 \|\| separation of finitely many disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given a finite collection of pairwise disjoint [[closed subset]]s in the topological space, there are pairwise disjoint open sets containing them. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any finite collection of pairwise disjoint closed subsets <math>A_1,A_2,\dots,A_n \subseteq X</math>, there exist pairwise disjoint open subsets <math>U_1,U_2, \dots, U_n</math> of <math>X</math> such that <math>A_i \subseteq U_i</math> for all <math>i</math>.		\| 4 \|\| separation of finitely many disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given a finite collection of pairwise disjoint [[closed subset]]s in the topological space, there are pairwise disjoint open sets containing them. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any finite collection of pairwise disjoint closed subsets <math>A_1,A_2,\dots,A_n \subseteq X</math>, there exist pairwise disjoint open subsets <math>U_1,U_2, \dots, U_n</math> of <math>X</math> such that <math>A_i \subseteq U_i</math> for all <math>i</math>.
	\|}		\|}

			Note that in each of the definitions, the [[T1 space]] assumption (that points are closed) can be replaced by the (a priori stronger) [[Hausdorff space]] assumption, without changing the meaning of the overall definition.

	Some versions of the definition omit the condition that points are closed subsets (in the above definition, the initial clause ''all points are closed'' would be removed from ''each'' of the above definitions). This is a ''different'', and ''weaker'', notion, and is covered here as [[normal-minus-Hausdorff space]].		Some versions of the definition omit the condition that points are closed subsets (in the above definition, the initial clause ''all points are closed'' would be removed from ''each'' of the above definitions). This is a ''different'', and ''weaker'', notion, and is covered here as [[normal-minus-Hausdorff space]].

Vipul at 21:30, 24 January 2012

2012-01-24T21:30:53Z

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	\|-		\|-
	\| 3 \|\| point-finite open cover has shrinking \|\| all points are closed, ''and'' every [[defining ingredient::point-finite collection\|point-finite]] [[open cover]] possesses a [[shrinking]]. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.		\| 3 \|\| point-finite open cover has shrinking \|\| all points are closed, ''and'' every [[defining ingredient::point-finite collection\|point-finite]] [[open cover]] possesses a [[shrinking]]. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.
			\|-
			\| 4 \|\| separation of finitely many disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given a finite collection of pairwise disjoint [[closed subset]]s in the topological space, there are pairwise disjoint open sets containing them. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any finite collection of pairwise disjoint closed subsets <math>A_1,A_2,\dots,A_n \subseteq X</math>, there exist pairwise disjoint open subsets <math>U_1,U_2, \dots, U_n</math> of <math>X</math> such that <math>A_i \subseteq U_i</math> for all <math>i</math>.
	\|}		\|}

Vipul at 21:01, 18 June 2011

2011-06-18T21:01:26Z

← Older revision		Revision as of 21:01, 18 June 2011
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	\| 1 \|\| separation of disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given any two disjoint [[closed subset]]s in the topological space, there are disjoint open sets containing them. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any two closed subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, B \subseteq V</math>, and <math>U \cap V = \varnothing</math>.		\| 1 \|\| separation of disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given any two disjoint [[closed subset]]s in the topological space, there are disjoint open sets containing them. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any two closed subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, B \subseteq V</math>, and <math>U \cap V = \varnothing</math>.
	\|-		\|-
	\| 2 \|\| separation of disjoint closed subsets by continuous functions \|\| all points in it are closed sets, and given any two disjoint closed subsets, there is a continuous function taking the value <math>0</math> at one closed set and 1 at the other. \|\| For all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any two closed subsets <math>A,B \subseteq X</math>, such that <math>A \cap B = \varnothing</math>, there exists a [[continuous map]] <math>f:X \to [0,1]</math> (to the [[closed unit interval]]) such that <math>f(x) = 0 \ \forall x \in A</math> and <math>f(x) = 1 \ \forall \ x \in B</math>.		\| 2 \|\| separation of disjoint closed subsets by continuous functions \|\| all points in it are closed sets, ''and'' given any two disjoint closed subsets, there is a continuous function taking the value <math>0</math> at one closed set and 1 at the other. \|\| For all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any two closed subsets <math>A,B \subseteq X</math>, such that <math>A \cap B = \varnothing</math>, there exists a [[continuous map]] <math>f:X \to [0,1]</math> (to the [[closed unit interval]]) such that <math>f(x) = 0 \ \forall x \in A</math> and <math>f(x) = 1 \ \forall \ x \in B</math>.
	\|-		\|-
	\| 3 \|\| point-finite open cover has shrinking \|\| all points are closed, and every [[defining ingredient::point-finite collection\|point-finite]] [[open cover]] possesses a [[shrinking]]. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.		\| 3 \|\| point-finite open cover has shrinking \|\| all points are closed, ''and'' every [[defining ingredient::point-finite collection\|point-finite]] [[open cover]] possesses a [[shrinking]]. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.
	\|}		\|}

Vipul at 21:00, 18 June 2011

2011-06-18T21:00:54Z

← Older revision		Revision as of 21:00, 18 June 2011
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	\| 2 \|\| separation of disjoint closed subsets by continuous functions \|\| all points in it are closed sets, and given any two disjoint closed subsets, there is a continuous function taking the value <math>0</math> at one closed set and 1 at the other. \|\| For all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any two closed subsets <math>A,B \subseteq X</math>, such that <math>A \cap B = \varnothing</math>, there exists a [[continuous map]] <math>f:X \to [0,1]</math> (to the [[closed unit interval]]) such that <math>f(x) = 0 \ \forall x \in A</math> and <math>f(x) = 1 \ \forall \ x \in B</math>.		\| 2 \|\| separation of disjoint closed subsets by continuous functions \|\| all points in it are closed sets, and given any two disjoint closed subsets, there is a continuous function taking the value <math>0</math> at one closed set and 1 at the other. \|\| For all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any two closed subsets <math>A,B \subseteq X</math>, such that <math>A \cap B = \varnothing</math>, there exists a [[continuous map]] <math>f:X \to [0,1]</math> (to the [[closed unit interval]]) such that <math>f(x) = 0 \ \forall x \in A</math> and <math>f(x) = 1 \ \forall \ x \in B</math>.
	\|-		\|-
	\| 3 \|\| point-finite open cover has shrinking \|\| all points are closed, and every [[defining ingredient::point-finite ~~cover~~\|point-finite]] [[open cover]] possesses a [[shrinking]]. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.		\| 3 \|\| point-finite open cover has shrinking \|\| all points are closed, and every [[defining ingredient::point-finite collection\|point-finite]] [[open cover]] possesses a [[shrinking]]. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.
	\|}		\|}

Vipul at 21:00, 18 June 2011

2011-06-18T21:00:27Z

← Older revision		Revision as of 21:00, 18 June 2011
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	! No. !! Shorthand !! A topological space is said to be normal if ... !! A topological space <math>X</math> is said to be normal if ...		! No. !! Shorthand !! A topological space is said to be normal if ... !! A topological space <math>X</math> is said to be normal if ...
	\|-		\|-
	\| 1 \|\| separation of disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given any two disjoint [[closed subset]]s in the topological space, there are disjoint open sets containing them. \|\| ~~every singleton subset~~ <math>\{ x \} ~~\subseteq X~~</math> is closed, ''and'' given any two closed subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, B \subseteq V</math>, and <math>U \cap V = \varnothing</math>.		\| 1 \|\| separation of disjoint closed subsets by open subsets \|\| all points in it are closed sets, ''and'' given any two disjoint [[closed subset]]s in the topological space, there are disjoint open sets containing them. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any two closed subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, B \subseteq V</math>, and <math>U \cap V = \varnothing</math>.
	\|-		\|-
	\| 2 \|\| separation of disjoint closed subsets by continuous functions \|\| all points in it are closed sets, and given any two disjoint closed subsets, there is a continuous function taking the value <math>0</math> at one closed set and 1 at the other. \|\| For all <math>x \in X</math>, the set <math>\{ x \}</math> is closed, and for any two closed subsets <math>A,B \subseteq X</math>, such that <math>A \cap B = \varnothing</math>, there exists a [[continuous map]] <math>f:X \to [0,1]</math> (to the [[closed unit interval]]) such that <math>f(x) = 0 \ \forall x \in A</math> and <math>f(x) = 1 \ \forall \ x \in B</math>.		\| 2 \|\| separation of disjoint closed subsets by continuous functions \|\| all points in it are closed sets, and given any two disjoint closed subsets, there is a continuous function taking the value <math>0</math> at one closed set and 1 at the other. \|\| For all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any two closed subsets <math>A,B \subseteq X</math>, such that <math>A \cap B = \varnothing</math>, there exists a [[continuous map]] <math>f:X \to [0,1]</math> (to the [[closed unit interval]]) such that <math>f(x) = 0 \ \forall x \in A</math> and <math>f(x) = 1 \ \forall \ x \in B</math>.
	\|-		\|-
	\| 3 \|\| point-finite open cover has shrinking \|\| all points are closed, and every [[defining ingredient::point-finite cover\|point-finite]] [[open cover]] possesses a [[shrinking]]. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed, and for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.		\| 3 \|\| point-finite open cover has shrinking \|\| all points are closed, and every [[defining ingredient::point-finite cover\|point-finite]] [[open cover]] possesses a [[shrinking]]. \|\| for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, and for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.
	\|}		\|}

Vipul at 20:59, 18 June 2011

2011-06-18T20:59:42Z

@@ Line 1: / Line 1: @@
 ==Definition==
-===Symbol-free definition===
+===Equivalent definitions in tabular format===
-A [[topological space]] is said to be '''normal''' if it satisfies the following equivalent conditions:
+{| class="sortable" border="1"
-# All points in it are closed sets, and given any two disjoint [[closed subset]]s in the topological space, there are disjoint open sets containing them.
+Some versions of the definition omit the condition that points are closed subsets (in the above definition, the initial clause ''all points are closed'' would be removed from ''each'' of the above definitions). This is a ''different'', and ''weaker'', notion, and is covered here as [[normal-minus-Hausdorff space]].
 ===Equivalence of definitions===

Vipul at 14:06, 15 September 2010

2010-09-15T14:06:17Z

@@ Line 1: / Line 1: @@
 ==Definition==
@@ Line 30: / Line 25: @@
 The direction (1) implies (2) follows from [[Urysohn's lemma]].
 ==Relation with other properties==

Vipul: /* Metaproperties */

2010-09-15T14:04:53Z

Metaproperties

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	==Metaproperties==		==Metaproperties==

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	!Metaproperty name !! Satisfied? !! Proof		!Metaproperty name !! Satisfied? !! Proof
	\|-		\|-