Talk:Metric space

From Wikipedia, the free encyclopedia

/Archive1

Contents 1 Split metric space into metric (mathematics) and metric space 2 Isometry 3 Homeomorphism and continuous functions 4 Is this an error? 5 british rail metric not a metric? 6 Superfluous condition?

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

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

• metric space article has become to long
• norm (mathematics) and normed vector space are separate articles too
• different metrics can be used to define the same metric space (depending of course on how one defines sameness)

MathMartin 12:28, 8 Apr 2005 (UTC)

These used to be separate and then got merged. Now they are separate again. I'm not going to complain too loud, but someone might. One could argue that norm (mathematics) and normed vector space should be merged. A topology is defined on the topological space page. -- Fropuff 16:19, 2005 Apr 8 (UTC)

To be honest I am not completely sure my split was a good thing. I have looked in the history but could not find the merge. Do you know the exact date ?

Norm (mathematics) and normed vector space should not be merged. A Frechet space can be defined using a countable collections of semi-norm (which should be merged with norm (mathematics)) so in this case it certainly makes sense to only talk about the norm and not the normed vector space.

My point is if your are talking about a topological vector space (a mixed structure) it is clearer to say the space is endowed with a norm and a metric than to say the space is a metric space and a normed vector space. For example it is clearer to say the space has a topology induced by a translation invariant metric than to say the space is a metric space with a translation invariant metric.

I do not care strongly about this split but I would like to get some other opinions before I merge the pages again.MathMartin 17:27, 8 Apr 2005 (UTC)

I think a split is good, as soon as a "critical mass" is reached, provided that sufficiently tight links are maintained, in order to avoid duplication of material, especially examples.

Concerning the concrete case, I think it is good to have the page separated into "norm" which could allow (more algebraic) discussions about norms (as functions) (definitions, constructions (product spaces, inner products,...)...) and their properties (equations, inequalities,...), while "normed space" could focus more on "global properties", remarkable subsets, and examples of such spaces. — MFH: Talk 19:36, 2 May 2005 (UTC)

I just learned about gage spaces, uniform spaces where the topology is defined by a family of pseudometrics. So I think the split in metric (mathematics) and metric space was reasonable.

[edit] Isometry

Isn't an isometry bijective? It is defined this way in Wikipedia. If so, there can be some simplification here. --JahJah 08:55, 21 September 2005 (UTC)

In my experience usually yes. However the usage is not completely standard. Looking at the books on my shelf gives the following:

• Steven Willard, General Topology (1970) defines an isometry as an order-preserving bijective (1-1 and onto) map. Defining two metrics to be isometric if there is an isometry between them.
• Eduard Čhec, Point Sets (1969) defines an isometry as an order-preserving injection (1-1), but says two metric spaces are isometric if there is a surjective (onto) isometry between them.
• Steen and Steenbach, Counterexamples in Topology, doesn't use the term, but defines isometric the same as the others.

I expect that Willard's and our definition at Isometry is the more standard (and modern?) especially since everyone defines isometric in the same way. If no one objects I would suggest changing this article to conform to the definition at Isometry. Paul August ☎ 15:47, 21 September 2005 (UTC)

I've checked a few references, and there is a certain amount of disagreement: however, for consistency I suggest changing this article to reflect Isometry. --JahJah 08:58, 22 September 2005 (UTC)

I would tend to agree. But in any case we should mention the varied usage at Isometry. Paul August ☎ 16:39, 22 September 2005 (UTC)

I'd call the "into" notion an "isometric embedding", though where clear from context it could slip to "isometry". BTW there's something a little redundant-sounding in the definition at isometry -- a "distance-preserving isomorphism"? What would be a non-distance-preserving isomorphism? A metric space doesn't have any structure that can't be recovered from the distance function. --Trovatore 17:00, 22 September 2005 (UTC)

I will raise this question at Wikipedia talk:WikiProject Mathematics. Paul August ☎ 16:32, 22 September 2005 (UTC)

Is this really a conflict? For an isometry in Euclidean geometry all of the plane must be in the image and pre-image; but in more general contexts we may want to consider, say, a ball within Euclidean 3-space. We need some way to talk about the common metric, and isometry seems a plausible word. It's quixotic for Wikipedia to try to enforce consistency when mathematics itself does not. Context is everything, so long as each is careful with its definitions. The problem is, Wikipedia is context free, so inconsistency must occur if Wikipedia is to be complete. (Hmm; that sounds eerily familiar.) Personally, I'm inclined towards Trovatore's distinction between isometry and isometric [map], insisting that isometric spaces be related by an isometric map that is further required to be bijective (and hence an isometry). --KSmrq^T 18:06, 22 September 2005 (UTC)

By the way, PlanetMath's Isometry and MathWorld's isometry also require an isometry to be a bijection. JahJah, would you please say which references you found which do not? Paul August ☎ 16:47, 22 September 2005 (UTC)

Munkres, Topology (a standard undergraduate/beginning graduate reference) defines isometry as distance-preserving but not onto. However, this is only in an exercise: the book concentrates on equivalence of metrics. --JahJah 07:38, 24 September 2005 (UTC)

The word "iso" also shows up in "isomorphism" where it is a bijective morphism. According to webster, the word comes from Greek "isos" meaning "equal". Ultimately I guess we can make it a convention that in this encyclopedia "isometry" will mean bijective distance-preserving map. This will be our ISO standard! Oleg Alexandrov 22:36, 22 September 2005 (UTC)

I prefer the names isometric map and isometric embedding for the 1-1 case, and isometry for the bijective case. While we are on the subject of conventions, I am not content with the usage of the term compact on WP for spaces which are not Hausdorff. The term quasi-compact was created for this reason (compact not-necessarily Hausdorff spaces); compact is reserved for Hausdorff spaces only. This is the standard in algebraic geometry as far as I can tell, and I feel that it is an important distinction. Whatever we decide for isometries should also end up here: Wikipedia:WikiProject_Mathematics/Conventions. - Gauge 00:29, 24 September 2005 (UTC)

I'm not a topologist, but that's not what I recall either as definition or usage. I have certainly seen the phrase compact Hausdorff space often enough to doubt it is redundant. What nationality are your textbooks? Septentrionalis 18:22, 24 September 2005 (UTC)

Bourbaki requires compact spaces to be Hausdorff, but that is not standard. For example, Steen and Seebach, and Willard don't require Hausdorff. Paul August ☎ 04:34, 25 September 2005 (UTC)

[edit] Homeomorphism and continuous functions

A metric structure on a given set induces a unique topological structure. This means that the notions of continuous function and homeomorphisms should be defined on the metric structure without any reference to topology. I think someone should rewrite that part of the article accordingly. Tomo 10:40, 30 October 2005 (UTC)

[edit] Is this an error?

Right now the article reads:

A metric space M is called bounded if there exists some number r, such that d(x,y) ≤ r for all x and y in M. The smallest possible such r is called the diameter of M.'

Is it the 'smallest' or 'largest'? Just trying to understand better. Plowboylifestyle 20:20, 29 December 2005 (UTC)

There is no mistake. The idea is that if you can squeeze the whole metric space into a ball of radius 4, you could even easier have it into a ball of radius 200, which is larger. Then, what you care about, the the smallest ball which still contains the whole space, and that's called the diameter. I hope that answers your question. Oleg Alexandrov (talk) 20:27, 29 December 2005 (UTC)

That's the problem with Wikipedia. When you don't understand something at first, you think it might be a typo. The math pages are pretty good though. Plowboylifestyle 02:52, 30 December 2005 (UTC)

[edit] british rail metric not a metric?

Does the British rail metric satisfy d(x,y)=0 iff x=y? I can only see this property happening only for the origin. I don't think it is a metric, but it *is* a norm on the cartesian product of the normed space in question:

||(x,y)|| = |x| + |y|

and hence can be turned into a metric on the cartesian product space by

d((x,y), (z,w)) = |x - z| + |y - w|

Any thoughts?

Seems to have been fixed since this question was asked. 82.42.16.20 00:47, 9 March 2006 (UTC)

In France (and French speaking countries) this is called "métrique SNCF" - would this merit being added on the main page ? — MFH:Talk 04:59, 10 March 2006 (UTC)

[edit] Superfluous condition?

Is the first condition really a definition condition for a metric?

It seems the non-negativity is not a condition, but rather a property of a distance function. Take any two points, A and B, and apply the triangle inequality to the A–A distance:

d(A, A) ≤ d(A, B) + d(B, A)

By the identity condition on the left side and the symmetry on the right side we get:

0 ≤ 2 × d(A, B)

Divide by two, and here is the result: d(A, B) ≥ 0.

Of course the non-negativity condition is necessary in quasimetric space, which does not guarantee d(x,y)=d(y,x), or in semimetric space, which does not guarantee the triangle inequality. But in the metric space it seems superfluous.
--CiaPan 22:44, 3 July 2006 (UTC)

Well, you're right I guess, but most textbooks etc. on metric spaces do in fact state the first condition as a condition rather than a property. So maybe we should follow convention and leave it the way it is? --HellFire 13:49, 18 July 2006 (UTC)

I would agree. I recall some author (a better mathematician than me) writing something to the effect (in this or a similar situation), that although it is possible to create a simpler set of axioms / conditions, the benefit is minimal, and the standard set have the advantage of greater clarity. Madmath789 21:04, 18 July 2006 (UTC)

I'm no great mathematician, but I think that there is a great benefit in separating conditions and properties: If you are trying to prove that a function is a distance function, the proof should have exactly as many sections as there are conditions; if you are trying to use a distance function to write a proof, then you might benefit by reviewing both the conditinos and properties. I understand moving some properties into the conditions to give a reader a more intuitive feel of something, but my personal feeling is that the logical structure of Conditions and Properties is worth more -- definitely though, I'm all for putting positive definiteness as the first derived property. (Jenny Harrison says something similar in Talk:Norm (mathematics).) If the inessential conditions are left in, I suggest we add a note saying that they are not actually conditions, but can be derived from the other conditions, that way you have the clarity that you (madmath, hellfire) are after but accuracy as well. MisterSheik 19:20, 21 July 2006 (UTC)

The positity is not inessential. I would suggest you cite a book or two where the positivity axiom is left out. As far as I am aware, it is always in. Oleg Alexandrov (talk) 06:53, 22 July 2006 (UTC)

I have added a note to the effect that (1) is not necessarily always part of the definition, together with a reference to a text that mentions it merely as a property. Hammerite 21:46, 24 July 2006 (UTC)

Thanks. I shortened it a bit, but agree that this needs be said. Oleg Alexandrov (talk) 03:05, 25 July 2006 (UTC)