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Revision as of 19:36, 20 July 2004

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:

• ds = -cdt + a ( dr /(1 - k r)) + r (dθ + sinθ dφ) )

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.

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 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 is certainly totally bounded. Note that open balls of 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 ≥ and ≤ 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)