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| In [[mathematics]], the concept of a '''generalised metric''' is a generalisation of that of a [[metric space|metric]], in which the distance is not a [[real number]] but taken from an arbitrary [[ordered field]].
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| In general, when we define [[metric space]] the distance function is taken to be a real-valued [[function (mathematics)|function]]. The real numbers form an ordered field which is [[archimedean property|Archimedean]] and [[complete ordered field|order complete]]. These metric spaces have some nice properties like: in a metric space [[compactness]], [[sequential compactness]] and [[countable compactness]] are equivalent etc. These properties may not, however, hold so easily if the distance function is taken in an arbitrary ordered field, instead of in <math>\scriptstyle \mathbb R</math>.
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| ==Preliminary definition==
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| Let <math>(F,+,\cdot,<)</math> be an arbitrary ordered field, and <math>M</math> a nonempty set; a function <math>d :M\times M\to F^+\cup\{0\}</math> is called a metric on <math>M</math>, iff the following conditions hold:
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| # <math>d(x,y)=0\Leftrightarrow x=y</math>;
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| # <math>d(x,y)=d(y,x)</math>, commutativity;
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| # <math>d(x,y)+d(y,z)\le d(x,z)</math>, triangle inequality.
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| It is not difficult to verify that the open balls <math>B(x,\delta)\;:=\{y\in M\;:d(x,y)<\delta\}</math> form a basis for a suitable topology, the latter called the ''[[metric topology]]'' on <math>M</math>, with the metric in <math>F</math>.
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| In view of the fact that <math>F</math> in its [[order topology]] is [[monotonically normal]], we would expect <math>M</math> to be at least [[Regular space|regular]].
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| == Further properties ==
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| However, under [[axiom of choice]], every general metric is [[monotonically normal]], for, given <math>x\in G</math>, where <math>G</math> is open, there is an open ball <math>B(x,\delta)</math> such that <math>x\in B(x,\delta)\subseteq G</math>. Take <math>\mu(x,G)=B(x,\delta/2)</math>. Verify the conditions for Monotone Normality.
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| The matter of wonder is that, even without choice, general metrics are [[monotonically normal]].
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| ''proof''.
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| Case I: ''F'' is an [[Archimedean field]].
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| Now, if ''x'' in <math>G, G</math> open, we may take <math>\mu(x,G):= B(x,1/2n(x,G))</math>, where <math>n(x,G):= \min\{n\in\mathbb N:B(x,1/n)\subseteq G\}</math>, and the trick is done without choice.
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| Case II: F is a non-Archimedean field.
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| For given <math>x\in G</math> where ''G'' is open, consider the set
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| <math>A(x,G):=\{a\in F\colon \forall n\in\mathbb N,B(x,n\cdot a)\subseteq G\}</math>.
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| The set ''A''(''x'', ''G'') is non-empty. For, as ''G'' is open, there is an open ball ''B''(''x'', ''k'') within ''G''. Now, as ''F'' is non-Archimdedean, <math>\mathbb N_F</math> is not bounded above, hence there is some <math>\xi\in F</math> with <math>\forall n\in\mathbb N\colon n\cdot 1\le\xi</math>. Putting <math>a=k\cdot (2\xi)^{-1}</math>, we see that <math>a</math> is in ''A''(''x'', ''G''). | |
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| Now define <math>\mu(x,G)=\bigcup\{B(x,a)\colon a\in A(x,G)\}</math>. We would show that with respect to this mu operator, the space is monotonically normal. Note that <math>\mu(x,G)\subseteq G</math>.
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| If ''y'' is not in ''G''(open set containing ''x'') and ''x'' is not in ''H''(open set containing ''y''), then we'd show that <math>\mu(x,G)\cap\mu(y,H)</math> is empty. If not, say ''z'' is in the intersection. Then
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| : <math>\exists a\in A(x,G)\colon d(x,z)<a;\;\;
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| \exists b\in A(y,H)\colon d(z,y)<b</math>.
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| From the above, we get that <math>d(x,y)\le d(x,z)+d(z,y)<2\cdot\max\{a,b\}</math>, which is impossible since this would imply that either ''y'' belongs to <math>\mu(x,G)\subseteq G</math> or ''x'' belongs to <math>\mu(y,H)\subseteq H</math>.
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| So we are done!
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| ==Discussion and links==
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| * Carlos R. Borges, ''A study of monotonically normal spaces'', Proceedings of the American Mathematical Society, Vol. 38, No. 1. (Mar., 1973), pp. 211–214. [http://links.jstor.org/sici?sici=0002-9939(197303)38%3A1%3C211%3AASOMNS%3E2.0.CO%3B2-8]
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| * FOM discussion [http://www.cs.nyu.edu/pipermail/fom/2007-August/011814.html link]
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| [[Category:Topology]]
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| [[Category:Norms (mathematics)]]
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| [[Category:Metric geometry]]
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