en>EmilJ: /* Construction of metric outer measures */ This is not a premeasure as we define it

2012-10-22T16:13:46Z

Construction of metric outer measures: This is not a premeasure as we define it

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	~~Hello friend~~. ~~Let me introduce myself~~. ~~I am Luther Aubrey~~. ~~Climbing~~ is ~~what adore performing~~. ~~My home~~ is ~~now in Kansas~~. ~~Interviewing~~ is ~~how he supports his family members but his marketing never arrives~~.<br><br>~~my weblog~~: [http://~~Www~~.~~Isoevent~~.fr/en/~~node~~/~~986034 Www~~.~~Isoevent~~.fr]		In mathematics, a '''monotonically normal space''' is a particular kind of normal space, with some special characteristics, and is such that it is hereditarily normal, and any two separated subsets are strongly separated. They are defined in terms of a monotone normality operator.

			A <math>T_1</math> topological space <math>(X,\mathcal T)</math> is said to be ''monotonically normal'' if the following condition holds:

			For every <math>x\in G</math>, where G is open, there is an open set <math>\mu(x,G)</math> such that
			#<math>x\in\mu(x,G)\subseteq G</math>
			#if <math>\mu(x,G)\cap\mu(y,H)\neq\emptyset</math> then either <math>x\in H</math> or <math>y\in G</math>.

			There are some equivalent criteria of monotone normality.

			==Equivalent definitions==
			===Definition 2===

			A space X is called monotonically normal if it is <math>T_1</math> and for each pair of disjoint closed subsets <math>A,B</math> there is an open set <math>G(A,B)</math> with the properties

			# <math>A\subseteq G(A,B)\subseteq G(A,B)^{-}\subseteq X\backslash B</math> and
			# <math>G(A,B)\subseteq G(A',B')</math>, whenever <math>A \subseteq A'</math> and <math>B ' \subseteq B</math>.

			This operator <math>G</math> is called '''monotone normality operator'''.

			Note that if G is a monotone normality operator, then <math>\tilde G</math> defined by <math>\tilde G(A,B)=G(A,B)\backslash G(B,A)^{-}</math> is also a monotone normality operator; and <math>\tilde G</math> satisfies
			:<math>\begin{align} \tilde G(A,B)\cap \tilde G(B,A)=\emptyset
			\end{align}</math>

			For this reason we some time take the monotone normality operator so as to satisfy the above requirement; and that facilitates the proof of some theorems and of the equivalence of the definitions as well.

			===Definition 3===

			A space X is called monotonically normal if it is <math>T_1</math>,and to each pair (A, B) of subsets of X, with <math>A\cap B^{-}=\emptyset=B\cap A^{-}</math>, one can assign an open subset G(A, B) of X such that
			# <math>A \subseteq G ( A , B) \subseteq G ( A , B)^{-}\subseteq X\backslash B,</math>
			# <math>G(A, B) \subseteq G(A', B') \mbox{ whenever }A \subseteq A' \mbox{ and } B'\subseteq B</math>.

			===Definition 4===

			A space X is called monotonically normal if it is <math>T_1</math> and there is a function H that assigns to each ordered pair (p,C) where C is closed and p is without C, an open set H(p,C) satisfying:
			# <math>p\in H(p,C)\subseteq X\backslash C</math>
			# if D is closed and <math>p\not\in C\supseteq D</math> then <math>H(p,C)\subseteq H(p,D)</math>
			# if <math>p\neq q</math> are points in X, then <math>H(p,\{q\})\cap H(q,\{p\})=\emptyset</math>.

			==Properties==
			An important example of these spaces would be, assuming Axiom of Choice, the linearly ordered spaces; however, it really needs [[axiom of choice]] for an arbitrary linear order to be [[normal space\|normal]] (see van Douwen's paper). Any [[generalised metric]] is monotonically normal even without choice. An important property of monotonically normal spaces is that any two separated subsets are strongly separated there. Monotone normality is hereditary property and a monotonically normal space is always normal by the first condition of the second equivalent definition.

			We list up some of the properties :
			# A [[closed map]] preserves monotone normality.
			# A monotonically normal space is hereditarily [[collectionwise normal]].
			# Elastic spaces are monotonically normal.

			== Some discussion links ==
			# R. W. Heath; D. J. Lutzer; P. L. Zenor, ''Monotonically Normal Spaces'', Transactions of the American Mathematical Society, Vol. 178. (Apr., 1973), pp. 481-493.
			# Carlos R. Borges, ''A study of monotonically normal spaces'', Proceedings of the American Mathematical Society, Vol. 38, No. 1. (Mar., 1973), pp. 211-214.
			# Eric K. van Douwen, ''Horrors of Topology Without AC: A Nonnormal Orderable Space'', Proceedings of the American Mathematical Society, Vol. 95, No. 1. (Sep., 1985), pp. 101-105.
			# P. M. Gartside, ''Cardinal Invariants of Monotonically Normal Spaces'', can be found [http://at.yorku.ca/p/a/a/d/08.dvi here] in Topology Atlas.
			# Henno Brandsma's discussion about Monotone Normality in Topology Atlas can be viewed [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2003;task=show_msg;msg=0383.0001 here]


			[[Category:Topology]]
			[[Category:Separation axioms]]
			[[Category:Properties of topological spaces]]

en>JumpDiscont: removed "disjoint, nonempty" from definition

2011-06-04T01:13:22Z

removed "disjoint, nonempty" from definition

New page

Hello friend. Let me introduce myself. I am Luther Aubrey. Climbing is what adore performing. My home is now in Kansas. Interviewing is how he supports his family members but his marketing never arrives.<br><br>my weblog: [http://Www.Isoevent.fr/en/node/986034 Www.Isoevent.fr]

Metric outer measure - Revision history

en>EmilJ: /* Construction of metric outer measures */ This is not a premeasure as we define it

en>JumpDiscont: removed "disjoint, nonempty" from definition