Talk:Metric space

The definition of metric space on my system at times does not display the symbols >= and <= correctly , as I mentioned earlier. Somebody suggested that I edit the incorrect symbols, but when I did, they were changed back. What should I do?

   ---S.A.G.

Zundark: For people like me who know almost nothing about it, why did you change the definition as you just did? Is your definition correct and the prior one incorrect, or are the two equivalent, with you just preferring one over the other? If so, why are they equivalent? Thanks, SJK.

They are equivalent, since the open ball is defined to be a subset of the metric space. LC changed the definition yesterday for some reason. I left it alone at the time, but this morning I decided I prefer the original definition, so I changed it back. I just feel that using "subset" could be a bit misleading, considering that it cannot be a proper subset. --Zundark, 2001 Dec 30

While not as elegant, maybe it is easier to understand if we first define a subset of a metric space to be bounded if it is subset of a ball, and then define the whole space to be bounded if it is bounded as a subset of itself. (Boundedness of the whole space is IMHO not as commonly encountered as boundedness of subsets.) --AxelBoldt

---

I just removed the following:

Applications in science cosmology - the FLRW metric. The metric of the spacetime of the universe, according to the standard big bang model, which as of 2002 was required by and consistent with all experiments and observations of spacetime events both close and as far away as possible from the spacetime event called human civilisation, is generally called the FLRW metric, and can be written in a spherically symmetric form such as:

• ds² = -c²dt² + a² ( dr² /(1 - k r²)) + r² (dθ² + sin²θ dφ²) )

This should go into an article about pseudo Riemannian manifolds somewhere. It is not related to the mathematical concept of metric space. AxelBoldt 11:40 Aug 28, 2002 (PDT)

---

I changed the term super-metric to ultrametric, since the later is far more common (use any search engine and count the results). Also added a note on quasimetrics: dropping property 4 requires changing property 3. -- Markus, Oct 1, 2003.

Contents

1 Totally bounded 2 Analisis 3 correction in definition of a metric space 4 Question regarding symbol representation and login 5 Summary of Changes to section on "Further definitions and properties" 6 The trivial distance metric? 7 Kevin Bacon 8 Split metric space into metric (mathematics) and metric space

Totally bounded

"The space M is called totally bounded if for every r > 0 there exist finitely many open balls of radius r whose union equals M."

Shouldn't this be "whose union CONTAINS M"? It seems silly to say, for example, that [0,1] is not totally bounded.

There's no difference, because strict containment is impossible - M is the whole space, and therefore contains all the balls. (And [0,1] is certainly totally bounded. Note that open balls of [0,1] need not be open in R.) --Zundark 17:41, 24 Nov 2003 (UTC)

Analisis

I think this section should be removed, I did it once, someone also did it before, but it still appears...

Tosha 21:07, 24 Feb 2004 (UTC)

Do not know how to stop this guy.... (I think this part is completely useless)

Tosha 01:20, 25 Feb 2004 (UTC)

I did it again...

Tosha 04:50, 25 Feb 2004 (UTC)

(It is clear that he is trying to do best, but does not know how...)

Tosha 04:53, 25 Feb 2004 (UTC)

correction in definition of a metric space

Perhaps there was a bug in the version of TEX you used but the definition of metric space should be re-written so that d(x,y) >= 0 always and also

d(x,z) + d(z,y) >= d(x,y) for all x,y,and z in the metric space.

    S. A. G.

That's what it says already (and it doesn't use TeX). Presumably the &ge; and &le; don't work on your browser - you can change them if you like. --Zundark 19:15, 18 Apr 2004 (UTC)

Question regarding symbol representation and login

With regard to my comment the other day about the definition of metric space ,I am using Internet Explorer version 6,and AOL 8 Plus and am enabling cookies for this site, but for some reason the &ge and &le seem to translate as a box symbol or solid circle sometimes.

Can someone help me with regard to this issue? Should I modify my preferences?

S. A. G.

Summary of Changes to section on "Further definitions and properties"

I have attempted to organise the section on "Further definitions and properties", which previously read like "other stuff", with a list of facts - useful reference for a mathematician but not much use to someone ploughing through for the first time. I have tried to add a bit more glue and organise things a bit more logically without stirring it up too much, or rewriting it from scratch.

1. added subheadings
2. expanded on totally bounded, adding context, making the "it can be shown that" a bit more specific without (I hope) getting too bogged down in details
3. slight reordering of chunks
4. added context for distance point to closed set. I put it in the "separation" subsection - it is a concrete demonstration of complete regularity.
5. added context for isometries - the reason these things are done is usually to get some embedding happening!
6. tweaks

I think it is an improvement. YMMV -- Andrew Kepert 07:49, 15 Jul 2004 (UTC)

The trivial distance metric?

The example hear called "The trivial distance metric", is (or was?) usually called the "discrete metric" (for example see: discrete space). In fact I've never heard of any other name for it. I'm tempted to change it. However if "trivial distance metric" is a common synonym, then perhaps I should include both? Anyone? Paul August 19:36, 20 Jul 2004 (UTC)

I think you should change it, since it gives rise to the discrete topology and I've not heard of the trivial metric before either. Lupin 19:59, 20 Jul 2004 (UTC)

I have changed it. It was used twice in the page, once my doing. I used the terminology of the page for internal consistency, not because it is the one I would use. The only reason I thought "trivial metric" could be a better name is that "discrete" only makes sense in reference to the topology, a more advanced concept. --Andrew Kepert 02:42, 22 Jul 2004 (UTC)

Andrew: Good. I think "discrete" is best, since it is, in my experience, the common usage. And despite the "trivial " nature of the metric, it really is a quite important one ;-). I will go ahead and add the link to discrete space Paul August 03:32, Jul 22, 2004 (UTC)

Another reason for my original reluctance is that I think more care is required here. Several metrics give the discrete topology (e.g. the |x-y| metric on the integers), so labelling this as the discrete metric is not entirely accurate. However if it is common usage, then leave it. (I can't recall this ever being given a name in my undergrad education, and haven't ever taught a metric space course, so I have never had a reason for giving it a name.) --Andrew Kepert 08:54, 23 Jul 2004 (UTC)

Update: I have an old copy of the "General Topology" Schaum outline by Seymour Lipschutz who says (page 111) that this "is usually called the trivial metric on X". I have a copy of Bourbaki's General topology somewhere .... and the only reference I can find (with my dodgy French) is in Section 3.8 of the first book which doesn't name it. Typically for Bourbaki it is only in the context of uniform structures.--Andrew Kepert 09:08, 23 Jul 2004 (UTC)

Looking through my (not necessarily representative ;-) texts, I find that Willard's "General Topology" (2.2 e. pp 17), Steen's "Counter Examples in Topology" (II.3.6 pp 41) and Goldberg's "Methods of Real Analysis" (4.2C 2) all use the term "discrete metric". A cursory examination of my english translation of Bourbaki's "General Topology", failed to find any mention of metric space. However it does name the analogous uniformity and uniform space, "discrete" (II.1.1 "examples of uniformities" pp 171). Furthermore, calling this metric the "trivial metric" could lead to confusion, since the so-called "indiscrete" topology (the only open sets are the empty set and the whole space) is also commonly called the "trivial topology", and since the pseudometric ( a pseudometric is defined by dropping the condition that d(x,y) = 0 => x = y) where d(x,y) = 0 for all x and y, is usually called the "trivial pseudometric". Finally, the notions of discete space (all points are far apart) and indiscrete space (all points are close together - notice the double entendre?) are, in a sense made precise in category theory "dual", so it's nice that the names used reflect this fact. ;-) Paul August 17:38, Jul 23, 2004 (UTC)

Fair enough - leave it as "discrete". --Andrew Kepert 03:26, 25 Jul 2004 (UTC)

Kevin Bacon

I removed the following from the example section

• The Kevin Bacon game defines a valid metric on the set of actors; however, the infinity of d(Kevin Bacon, Fred Ott) means that the metric space is not complete.

If d(Kevin Bacon, Fred Ott) = ∞, then this is not a metric according to the definition in the article, which states that d maps to the reals (of course, this can be fixed). Anyway, I don't understand why the space is not complete; I would have thought that it is a discrete space and hence complete. -- Jitse Niesen 12:56, 3 Sep 2004 (UTC)

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.