Talk:Metric (mathematics)

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[edit] Split metric space into metric (mathematics) and metric space

I split metric space into metric (mathematics) and metric space. Reasons were

MathMartin 12:27, 8 Apr 2005 (UTC)

I think this splitting up is a good idea, but more cross-links are needed between these two pages (and they should refer to precise sections and/or examples, so section titles should be well-thought).
I would only criticize a bit your last point: In mathematics, "the same" should mean equal, IMHO. "Equivalent metrics" are now defined, they define the same topology, i.e. the same topological space, but not the same metric space (defined as a certain 2-tuple).

— MFH: Talk 13:28, 19 Apr 2005 (UTC)

Section linking between articles is a very bad idea, because the links are prone to breaking. If section linking cannot be avoided the two articles should be merged again. What do you mean by same should mean equal ? MathMartin 13:44, 19 Apr 2005 (UTC)

[edit] bug fixes and cleanup suggested

  1. in "Definition", it is written "function function" - should we just delete the first "function", or was it intended to replace it with "distance" ?
  2. I suggest that at least "pseudo" and "ultra" should be defined in the same place than "intrinsic" (much more used, IMHO), i.e. in "Definition"
  3. "Notes" (I would prefer "Comments") - here the last phrase (2,3,5 => 1) refers to a property "5" which is not defined ; please correct (and, if possible in 1 or 2 lines, write the proof).
  4. The notion of 'metric induced by a norm' is already used 2 sections before(Examples) so there should be at least a forward reference (maybe better give the definition in a subsection of "examples")

— MFH: Talk 12:59, 19 Apr 2005 (UTC)

I recently split this article from metric space so most of the bugs are probably from me. I had intended to do a cleanup in the next few days. All your suggestions seem reasonable so go ahead. However I would keep the old section title Notes as it is more common in math articles on wikipedia.

As a sidenote, I think it is uneccessary to discuss such small modifications on the talk page. The next time you should be more bold and just do the changes. MathMartin 13:27, 19 Apr 2005 (UTC)

[edit] remove fraction

I changed

da( x, y ) = Σ ai pi(x-y) / (1 + pi(x,y))

to

d(x,y)=\sum_{n=1}^\infty \frac{1}{2^n} \frac{p_n(x-y)}{1+p_n(x-y)}

so I changed that ai coefficent to a specific geometric series. That's what's in my book, and I guess it's needed to ensure convergence of this sum -Lethe | Talk 05:49, July 11, 2005 (UTC)

[edit] Content syncronizing

The articles Metric (mathematics), Distance, Metric space, and maybe something else, have evolved independently after mathMarti decided to split them. May be his intentions were good, but now the articles create a confusing mess rather than a coherent mathematical discourse. Please someone can initiate a microproject to put some order/system into this topic? `'Míkka 17:48, 8 October 2007 (UTC)

[edit] Generalized metrics

I have made several changes in metric and metric space. Most of them should speak for themselves, but perhaps I should explain why I have removed all references to hemimetric and prametric. Most people are reluctant to remove something just because they have never heard of it. So am I, but this was a special case.

  • The only reference given for prametric is Arkhangelskii and Pontryagin: General Topology I. My library doesn't have the book, and it's not on Google books either, therefore I can't check that the strange word "prametric" really appears in the book. Google has 1660 hits for it. Each of the first 100 hits is either an obvious misspelling for "parametric", or it's on Wikipedia or a clone. Google Scholar has 94 hits, every single one is an obvious misspelling for "parametric". Google books has, surprise, one relevant hit among 25 misspellings: Ruben Aldrovandi and J. G. Pereira, An Introduction to Geometrical Physics. So that's a total of two books defining the term. (The book on Google Books doesn't seem to do anything with it except that it derives a topology.)
  • The only reference given for hemimetric is PlanetMath. PlanetMath does not give a reference at all. The presyllable "hemi" is usually used as a synonym for "semi". The concept itself is often referred to as pseudo-quasi-metric, which is a bit longer but logical once pseudometric and quasimetric have been defined. Google has 852 hits. Some of them are for "m-hemimetric", whatever that is, and I suspect that many are variants for "semimetric". Google Scholar has 0 hits. Google Books has 2 hits: One for "m-hemi-metric", and one for "quasi-hemi-metric", a combination that makes no sense with our definition.

The only frequently used terms for generalized metrics appear to be:

Since all combinations of the axioms can be expressed by combinations of pseudo-, semi- and quasi-, there is no reason why Wikipedia should coin a strange new term such as "prametric". (Or help the translator of a Russian book to do so.)

To be absolutely sure, I made a number of searches in publications. Numbers are for search without hypen/with hyphen, e.g. prametric/pra-metric.

word Math. Reviews Zentralblatt
prametric 0/0 0/0
prometric 0/0 1/0
premetric 21/14 12/15
hemimetric 2/3 1/0
pseudometric 821/311 386/254
semimetric 236/112 98/84
quasimetric 138/273 71/253

I think these results show clearly that it is seriously misleading to direct unsuspecting readers to the articles on prametric and hemimetric. The remaining question is whether to delete them altogether. I think there should be something like a notability criterion for definitions, and I suspect they wouldn't pass.

Perhaps some people who actually work in topology and functional analysis, or in related disciplines, can check whether the passages about the two definitions of semimetric really describe usage in their respective fields. I have also asked the MSC2010 group to clarify what the MSC2000 category "54E25 Semimetric spaces" is about. --Hans Adler (talk) 02:56, 22 November 2007 (UTC)

PS: I was drawn into this because preclosure operator was misspelled as praclosure operator and had a link to prametric space. "Praclosure" has 0 hits in Google Scholar and Google Books, but 167 on Google. All on Wikipedia, including clones and translations. --Hans Adler (talk) 03:01, 22 November 2007 (UTC)

[edit] Axioms of a metric

Correct me if I'm wrong but I believe that the first condition in the definition is redundant as it follows from identity of indiscernibles (condition 2) and triangle inequality (condition 4) Jergosh (talk) 20:36, 13 May 2008 (UTC)

A little update: I checked the definition with a book I'm using at my university (not a definitive resource as it is printed locally) and it states that there are three 'axioms of a metric' (identical to axioms 2, 3 and 4 here on wikipedia) and that nonnegativity follows from them (apparently all three). I will try to research this further but I have limited access to sources in English. Jergosh (talk) 20:52, 13 May 2008 (UTC)

It's easy to prove 1 from 2-4 (and an easy counterexample with two points shows just 2 and 4 are not enough). But this doesn't mean we need to drop 1. --Hans Adler (talk) 23:38, 13 May 2008 (UTC)
Why keep it if it's redundant? Perhaps it would be better style to keep 2-4 as axiom and remark that nonnegativity follows from them (i. e. is not an axiom but a useful property). Jergosh (talk) 06:41, 14 May 2008 (UTC)
It becomes important when you consider certain subsets of the axioms; for many such subsets there are names. Positive definiteness is even specifically mentioned in the article. Also in some axiomatisations non-negativity is built in even before the first axiom is mentioned, by saying that "a metric is a function d: X × X → R0+ satisfying..."
I am not fundamentally opposed, I just don't think it would be a clear improvement to drop the axiom. --Hans Adler (talk) 11:51, 14 May 2008 (UTC)