Talk:Metric space/Archive 2

31: 901:

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

3600:). But when I tried to post it, your post caused an edit conflict, and the screen that goes with edit conflicts is confusing and by then, I was sick of the whole thing. I did NOT want to spend yet more time on it, as I was already mad at myself for ignoring my homework for the entire evening. So I said "the hell with it", never intending to come back here. But I saw the "new message" line, was beguiled, and came anyway. 1078:

I need a quick definition of rap music terms like 'shizzle' or 'biotch', a definition written in more rap terms is of little help. Same with math -- nothing against the utility or beauty of such language taken a whole, but readers want "just the facts", not to learn secret handshakes or to join the tribe, and not because they don't like or respect the tribe, but because they haven't enough time. --

1263:

define a metric, usually a logarithm of cross-ratio. So all the models of the hyperbolic plane are isomorphic metric spaces, but to grasp a distance function some particular model must be in hand. Thus I recommend statement of the example in terms of a model; fortunately the editor left the anchored link to the model section of the hyperbolic geometry article.

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article I worked on or has an admin friend. But if you get banned, you can't edit again unless you get a new ISP. Three of my friends were treated shockingly unfairly and permanently banned for insisting on including material from a seminal article in a important academic journal. The "owner" just didn't like the information in it.

205:(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. 3917:

What I'm really about is that if imaginary distance is negative real distance, then it is absolutely, unquestionably trivial to calculate h-zero (rate that the universe expands) using only its age and c. No gravity enters into it. The answer thus predicted is the correct expansion rate to within the

3756:

You do not need to explain me the meaning and structure of the indefinite metric of the space-time; I understand it well enough (the last 50 years). But I cannot agree when you write "The rest of your post is about the meaning of words. But semiotics is both boring and irrelevant—particularly in math

3378:

the distance in space. That's because time is just another dimension, like x, y, and z. The ONLY difference between them is that distances in time are negative real distances. A yardstick pointing in the "time direction" would be -36 inches long. If you prefer, you can say that the time distance is a

1077:

Or perhaps you're saying that since math bywords exist, they must be tolerated (and learned) to read WP articles. Seems like the same argument could be made for all slang and jargon. No, the point of general encyclopedia is save time by minimizing the amount of new syntax and vocabulary needed. If

3904:

The reason I'm still here is that i've only been reading this stuff for 2 years and i wanted to see others' opinions about whether it is legitimate to view imaginary distance as negative real distance. Because in the spacetime interval metric (pyth theorm in 4D), it blatantly ACTS like neg distance.

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This is usually just asserted in special relativity books instead of being derived, but Einstein said so, and his derivation gives true insight into the nature of space and time. If you add ct² instead of subtract it, all kinds of other equations become internally inconsistent (i.e., they contradict

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Once again an editor has pruned the statement, "Any model of the hyperbolic plane is a metric space" to "The hyperbolic plane is a metric space." In the lives of Lobachevski and Bolyai there was a "hyperbolic plane" but only faintly a metric. It took the algebraic models of the hyperbolic plane to

1242:

In listing examples of metric spaces we elucidate the concept by drawing on a student's previous experience. It may be that a student of metric spaces has not been introduced to hyperbolic geometry. In that case, it is only reasonable to point to a model of the hyperbolic structure that is asserted

900:

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

710:

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

3752:

Well, I am sorry for the inconvenience caused by the edit conflict, but I do not feel guilty. and in fact, I also had such inconveniences, this is the life in WP. Also, I never appeal to arbcom (check it if you doubt), has no admin friends (well, this is hard to check, just believe me), and I will

3900:

I don't give a fchh about this article anymore. As I've said several times, I abandon articles when other people get excited and hostile about telling me I'm wrong . Let the fanatics own articles. Because of the corrupt, fu cked up management I've seen (twice), I don't really care very much about

3605:

Another reason I didn't reply is that I learned never to argue with "grownups," particularly on Knowledge. If someone takes you to arbcom, just abandon your wiki-name (cut your own wiki-throat) and create a new account. I'll do that at the drop of a hat if someone who disagrees with me "owns" an

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way to talk about the common metric, and isometry seems a plausible word. It's quixotic for Knowledge 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, Knowledge is context free, so inconsistency

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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

878:

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.

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and physics." Note that this discussion was not started by me. Other persons were disturbed with your formulation. And I understand, why. You may neglect "the meaning of words" if you like, but probably not when contributing to an encyclopedia. Otherwise you are at risk of being not understood.

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is important in relativity theory, and no doubt that this space is endowed with something that physicists routinely call "metric". Here is the question: do they call it "metric space"? I suspect that the term "metric space" is always (or almost always?) used according to the "mathematical"

960:

I am deleting the "more compact but equivalent" definition from the definition section, as it seems to be wrong. Consider the following example metric d(x,y) = y-x. This clearly fulfills d(x,x) = 0 and d(x,z) <= d(x,y)+ d(y,z). However, it is not a metric according to the "less compact"

625:

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.

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to exist. For this reason I have edited the example to guide the student to links for such models. To claim baldly that a hyperbolic structure is a metric space can only discourage an inquistor trying to build a new concept in mind. The notion is a bit subtle since we are dealing with a

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A "metric space" is something standard in mathematics that can be found in any number of elementary textbooks. It is clear that pseudo-Riemannian manifolds of indefinite signature (including the Lorentzian manifolds of relativity theory) are not "metric spaces" in the sense that

4002:

it's not about your age, and it's not about owning anything. It is, to some extent, about what to call things, because our readers rely on us to use standard terminology that corresponds to what is found in the standard literature. It doesn't matter whether it's the

3712:

We observe that both ct and ct² are positive. They're just like the x, y, and z terms. The Jedi mind trick happens when you have to SUBTRACT that ordinary distance from the squares of the other three distances. It is then that the time term becomes a negative real

905:.) 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. 1067:

means "having many ramifications"? First draft recast: "..by using a different metric we can construct non-Euclidean geometries with many ramifications..." Sounds a bit high tone, perhaps something better will come to mind, but at least it is more

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conceives of space in 3DE? It's arguable that some people don't -- the text seems patronizing. Safer to say "The most familiar space might be 3-dimensional Euclidean space", or even use a qualifier like "most familiar space to western society".

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But I do not understand you. A negative distance means a positive squared distance, right? A negative squared distance means an imaginary distance, right? (And an imaginary number cannot be "positive" or "negative", since the choice between

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Does anyone know how a function of this form might be called? It's essentially a generalization of isometry. (So, the map is injective and has closed range, assuming completeness.) It's similar to Lipschitz-ness, but not really the same. --

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I think "interesting" is more a byword for that which presents (or has presented) fruitful opportunities for mathematical investagation not found in every geometry. It's not necessarily merely a case of "I like" or "mathematicians like".

1884: 3852:, which claims that the two terms are synonyms. I took differential geometry from O'Neill, whose book is called "Semi-Riemannian Geometry", and I don't have it handy but I'm pretty sure it used the terminology as I have used it here. -- 1484: 1095:

Oh, no, I'm not trying to say that the obscurities of mathematics communication ought to be foisted on a wider audience without explanation. I was offering my understanding of its usage in order to help inform a rephrasing.

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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:

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I mean that for any manifold of metric signature (3,1), you increase the total distance by adding to any of the 3 (spatial) vectors, or by SUBTRACTING from the 1 imaginary-length vector (in this case, time). This is the

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link to the page on the identity of indiscernibles? The principle of identity of indiscernibles says that two objects have all the same properties iff they are equal, but this condition is only for a single property.

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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

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Call it what you want; I don't care. Call it a "fake" or "pseudo" metric if you prefer. But Einstein and every book on SR says that "the metric for spacetime is ." Paypal me five bucks and I'll look one up for you.

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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.

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I don't know if you've seen the metric of the manifold called "spacetime", but it's the Pythagorean theorem with an extra term (for time), and that term is SUBTRACTED from the sum of the other three squares. Per

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Nobody has ever noticed this before because no one ever thinks of time as negative spatial distance and because people automatically dismiss it if a 15-y/o (at the time this occurred to me) kid came up with it.

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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:

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Again, I'll trade you references on all this for 10 bux paypal. Then I can buy beer from the guys in the dorm, which I won't be able to buy at the store for many years. But it's either that or give public

2378:

I have fixed this error by requiring that the norm in question be the usual Eudlidean norm, and mentioning that there are other norms for which the normed product metric as defined is really a metric. -

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of a mathematical structure. Nevertheless, pointing to the model in the reference makes plain the path to assimilating an important instance of this metric space concept encountered in topological study.

3653:

That also preserves the integrity of the pythag theorem. And BTW, "c" is only there so we can refer to time as a spatial distance in the same units as x, y, and z. (e.g., light years instead of years).

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Also, Google search for "indefinite metric space" gives some occurrences of "indefinite-metric space" and in rare cases "indefinite metric space". Most authors hesitate to use the latter term

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is not some universally agreed upon property. Maybe the passage's author meant "mathematically interesting", but that adjective is vague as it could fairly apply to any category of math.--

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occur if Knowledge is to be complete. (Hmm; that sounds eerily familiar.) Personally, I'm inclined towards Trovatore's distinction between isometry and isometric , insisting that isometric

419:-- 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. -- 3566:

Thus, such remark as "In the most general definition of a metric space, the distance between set elements can be negative. Spaces like these are important in the theory of relativity."

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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

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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.

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mean by the term "metric space". An attempt to insist that they are, contradicting a host of reliable sources, is not appropriate for an encyclopedia. Also, I think Trovatore means

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I don't think it makes sense (e.g., the time will depend on the state of the CPU cache). In any case, there are no Google hits for the term, so it must be pretty obscure, probably

3213:

If you increase x and want the distance to remain unchanged, you have to subtract some y or z, or you can ADD an amount to t. This is exactly the situation for every point on the

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Wow, what a lot of verbiage over such a cut-and-dried thing. Semi-Riemannian manifolds (non-positve-definite ones) are very important. But they are not metric spaces. Done. --

3217:. Each of those points has zero distance from the center, but positive and negative distances from each other. That's right; 4D distance in a space with a (3,1) signature is 3067: 2962: 2711: 935:

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.

3496:

More on the terminology. Search for "indefinite metric" (on WP) gives many pages, but I never see "metric space" in that context. I see "indefinite inner product" in "

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On the metric page this property is also called the coincidence property. I would suggest changing the name "identity of indiscernibles" to "coincidence property".

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Before y'all got all worked up about whether a pseudometric is called a metric, it seemed like nobody has any real problem with seeing imaginary distance this way.

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that various great and lesser mathematicians and scientists have been and are interested in such geometries, and that these geometries have been useful. It's an

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Yes, yes; they're complex conjugates. What's your point? Unfortunately, you leave it out, so I'm not sure what it is. It might help to look at the metric again:

580: 537: 4011:, so that when users come across other works that discuss metric spaces, they will be able to map the concepts found in those works to the ones we discuss. 1741:{\displaystyle N(d_{1},...,d_{n}){\Big (}(x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n}){\Big )}=N{\Big (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\Big )}} 1761:

Unfortunately, this is not actually true, as was pointed out on Ben Green's Metric and Topological Spaces, Example Sheet 1, Easter 2008. Specifically, let

1388: 2652:). This is incompatible with the definition stated both within the article and in the diameter article; consider an equilateral triangle (or any regular 323:(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. 164:

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 ?

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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

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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.

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Before talking about math, I want to say that actually, I spent nearly three hours writing (what I think was) a response both jocular and eloquent. [

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I've checked a few references, and there is a certain amount of disagreement: however, for consistency I suggest changing this article to reflect

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Bourbaki requires compact spaces to be Hausdorff, but that is not standard. For example, Steen and Seebach, and Willard don't require Hausdorff.

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is deleted by Lovysinghal saying "Def of metrizable spaces isn't exactly the one given". Why? What was wrong with the definition? I restore it.

3778: 437: 4039:!!! You think I want to add my model to WP? NO! Astrophysical Journal, sure. But only if it's not crackpot, which is my greatest fear. Verdana 2078: 1125:. 2nd draft attempt: "..by using different metrics a wide and productive range of non-Euclidean geometries have been discovered..." -- 3945: 3760:

As for me, all math you wrote (above, not in the article) is OK, and still, the problem (posed by others and me too) is not resolved.

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that I only want to understand stuff, not what names people give the stuff. I never said word one about terminology on WP. --Verdana

3658:"ct" is expressed in positive units (like 4 light-years). The "negative" comes from the minus sign to the left of the squared value. 3517: 3729:

The rest of your post is about the meaning of words. But semiotics is both boring and irrelevant—particularly in math and physics.

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

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I agree that the "identity of indiscernibles" is irrelevant here. But nevertheless this comment is not true. The statement that

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seems to misrepresent the definition of diameter; by looking at the picture, one is lead to believe that the diameter of a set

3470:

A negative distance means a positive squared distance, right? A negative squared distance means an imaginary distance, right?

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not edit-war with you (or anyone); if we cannot agree then I rather wait for more participants, in order to get a consensus.

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A space with a (3,1) metric signature has one dimension with negative length relative to the other three, by definition.

2452: 1294: 2443:. I'm putting the proposal instead of boldly making the change because this may be discussed before. Any objection? -- 1897: 2003: 1950: 59: 2369:{\sqrt {8}}+{\sqrt {7}}=N(d_{1},d_{2}){\Big (}(0,0),(2,2){\Big )}+N(d_{1},d_{2}){\Big (}(2,2),(4,1){\Big )},}" /: --> 1043: 729:. When you don't understand something at first, you think it might be a typo. The math pages are pretty good though. 3849: 3820: 3532: 2079:{\sqrt {8}}+{\sqrt {7}}=N(d_{1},d_{2}){\Big (}(0,0),(2,2){\Big )}+N(d_{1},d_{2}){\Big (}(2,2),(4,1){\Big )},}": --> 1764: 592:

I'm not a topologist, but that's not what I recall either as definition or usage. I have certainly seen the phrase

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http://en.wikipedia.org/Talk:Metric_(mathematics)#Split_metric_space_into_metric_.28mathematics.29_and_metric_space

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I do not care strongly about this split but I would like to get some other opinions before I merge the pages again.

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also require an isometry to be a bijection. JahJah, would you please say which references you found which do not?

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being wrong. It's like a horrible pus-boil you don't know you have. If I have one on me, I want to know about it

3827:. Semi-Riemannian manifolds actually are metric spaces, but do not possess what would normally be regarded as a 3479:

definition. That is, terminologically, "your generalization" is a metric (on a space), but not a "metric space".

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for the bijective case. While we are on the subject of conventions, I am not content with the usage of the term

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Well, you're right I guess, but most textbooks etc. on metric spaces do in fact state the first condition as a

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Take it out of the lede then; I don't care. But you can't then say that distances are always positive. Verdana

3911:

If you still want to fight about what names to call stuff, fine 4 U. But do it w/o me because I don't care.

2363:{\sqrt {8}}+{\sqrt {7}}=N(d_{1},d_{2}){\Big (}(0,0),(2,2){\Big )}+N(d_{1},d_{2}){\Big (}(2,2),(4,1){\Big )},} 812:

of a distance function. Take any two points, A and B, and apply the triangle inequality to the A–A distance:

3868: 2497:! I misread and misinterpreted the written sentence as "... which arises in this way ..." instead of "which 902: 3253:

Define it however you want; that doesn't change the topological mathematics that rules the physics. Verdana

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You may neglect "the meaning of words" if you like, but probably not when contributing to an encyclopedia.

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positive imaginary distance instead of a negative real distance, but that's unnecessarily, uhh... complex.

3351: 2448: 2404: 1180:). Are there such things as a non-symmetric (non-commutative) metric spaces where this is not the case? — 966: 366:

in the same way. If no one objects I would suggest changing this article to conform to the definition at

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Note that the given matrix is symmetric and positive definite, so indeed defines an inner product; let

192:) so in this case it certainly makes sense to only talk about the norm and not the normed vector space. 4015:

Accck! You people just won't stop lecturing me about terminology, as if I'm arguing with you about it!

2903: 2502: 764: 3933: 3575: 3002: 2472: 946: 922: 802: 712: 545: 262: 95: 80: 3831:. These are different, but related concepts, that only agree in the case of Riemannian manifolds. 3136:

In the most general definition of a metric space, the distance between set elements can be negative.

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that in this encyclopedia "isometry" will mean bijective distance-preserving map. This will be our

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space" could focus more on "global properties", remarkable subsets, and examples of such spaces.

177: 146: 116: 839:, which does not guarantee the triangle inequality. But in the metric space it seems superfluous. 306:

bijective? It is defined this way in Knowledge. If so, there can be some simplification here. --

3466:

During 3 days you did not address my question above. Is it not visible enough? Here I repeat it:

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but I really doubt it is in the article's scope. Even if it is, it should rather be mentioned in

2645:{\displaystyle \inf\{2r\in \mathbf {R} :A\subset {\overline {B}}(x,r){\text{ for some }}x\in A\}} 1244: 1097: 1051: 287: 189: 173: 142: 112: 4106: 3808: 250: 3029: 2924: 4094: 3937: 3878: 3838: 3298:

of a metric space definition, which allow the distance between set elements be negative. (...)

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What group of people does the pronoun "our" encompass? Is it known for certain that everyone

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I am not sure exactly what you mean by imaginary distance acting like negative real distance

1879:{\displaystyle \langle x,y\rangle =x\cdot {\begin{bmatrix}3&-2\\-2&3\end{bmatrix}}y.} 532:

The word "iso" also shows up in "isomorphism" where it is a bijective morphism. According to

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Maybe you have a different usage of "semi-Riemannian" in mind? The link you gave points to

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disagrees with the definition of metric space given just a little bit lower in the article!

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could not imagine any other space but 3DE, and thus built some of his metaphysics on it.)

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Say, that's too bad! If I had any, I'd pay BIG MONEY for my model to be proven wrong. I

2836: 2810: 536:, the word comes from Greek "isos" meaning "equal". Ultimately I guess we can make it a 4098: 3853: 3800: 3505: 3295: 3268: 3208:

This is the metric for 4D distance, which is called the "interval" between two points.

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The negative squared distance will always be negative if the distance in time is : -->

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

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Say, that's 2 bad! Come back if you can ever prove it's not true. That's what I seek.

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Yeah. Good. I'll get you a front-row seat for my speech at the Nobel ceremony.Verdana

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this (WP) project anymore, either. I just like making things better (like grammar).

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Knowledge talk:WikiProject Mathematics#Another terminology problem: what is a metric?

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I hear you ask, "Why must you subtract the time distance from the sum of the others?"

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Just added suggestions for principles (and implicitly clean-up) to the discussion in

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the choice between +i and -i is only a convention; their properties are identical.)

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Yeah, my mom named me after a Windows font. My sister is Helvetica oblique. Verdana

3816: 3578:" or, if you want to emphasize it more, in the lead of that article, but not here. 3305: 1264: 1248: 1228: 1224: 1195: 1001: 266: 258: 154: 106: 99: 91: 84: 76: 4135: 3882: 3861: 3842: 3789: 3769: 3587: 3491: 3453: 3429: 3354:; there, in some sense, squared distance may be negative, not the distance itself. 3317: 3272: 3155:

I don't know the correct generalization that covers negative distances, but it is

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This may well be true, but that does not change the definition of metric space!)

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A topological space which can arise in this way from a metric space is called a

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If you wish to start a new discussion or revive an old one, please do so on the

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to X, Y. To the best of my knowledge, X, Y are much more commonly used than M

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I would tend to agree. But in any case we should mention the varied usage at

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And by the way, what is "important in the theory of relativity" is probably

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often enough to doubt it is redundant. What nationality are your textbooks?

331: 185: 2362:{\displaystyle N(d_{1},d_{2}){\Big (}(0,0),(4,1){\Big )}={\sqrt {35}}: --> 3563:

The former term indicates that this space is not quite a "metric space".

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on 19 April 2014, and it seems untrue to me, too. I think one could say

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

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was created for this reason (compact not-necessarily Hausdorff spaces);

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is twice the radius of the smallest ball that contains the set (i.e.,

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It's already confusing for people who don't understand it — like you.

861:. So maybe we should follow convention and leave it the way it is? -- 700:

Is it the 'smallest' or 'largest'? Just trying to understand better.

362:

is the more standard (and modern?) especially since everyone defines

351:, doesn't use the term, but defines isometric the same as the others. 4112:

Now, I'm gonna try reeeal hard not to come back to this page.Verdana

4085:

and I'm not going to say off the top of my head that it's not true.

533: 774:

In France (and French speaking countries) this is called "métrique

945:

Thanks. I shortened it a bit, but agree that this needs be said.

3678:

A negative squared distance means an imaginary distance, right?

3615:

A negative squared distance means an imaginary distance, right?

2807:, and it is clear that this is best (if the triangle is labeled 2733:, but the full triangle is contained in a closed ball of radius 775: 4035:

Original insight is a great thing, but not in an encyclopedia.

3440:

Well, I've deleted the confusing statement. It is enough that "

3690:

Yeah, only complex numbers can be. Thank you for educating me.

3670:

A negative distance means a positive squared distance, right?

3524:". But "indefinite metric space" would be a neologism, right? 3127:

Utterly untrue and confusing statement in introductory section

3103:

having some property the same. It is related only to x and y

541: 25: 988:

understanding of space is the 3-dimensional Euclidean space.

3243:

Spaces like these are important in the theory of relativity.

3686:

And an imaginary number cannot be "positive" or "negative"

3576:

Metric (mathematics)#Important cases of generalized metrics

2902:

Should we scrounge up a more pedagogically useful diagram?

808:

It seems the non-negativity is not a condition, but rather

3535:; there, such objects are never called "metric spaces". 763:

Seems to have been fixed since this question was asked.

4093:

The place to ask is at the mathematics reference desk,

3918:

accuracy with which we can measure the expansion rate.

3287: 1210: 831:

Of course the non-negativity condition is necessary in

3403:

is only a convention; their properties are identical.)

2007: 1954: 1901: 1833: 1768: 778:" - would this merit being added on the main page ? — 4020:

The closest thing i said was that names don't matter

3549:

A man eating shark is a man who is eating shark meat.

3077: 3032: 2972: 2927: 2885: 2865: 2839: 2813: 2793: 2740: 2719: 2693: 2658: 2570: 2550: 2530: 2082: 2006: 1953: 1900: 1803: 1767: 1499: 1391: 1297: 126:

metric space (depending of course on how one defines

828:

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

3203:

Einstein's spacetime metric: d² = x² + y² +z² - ct²

3189:confusing for someone trying to learn the concept. 1368:{\displaystyle (M_{1},d_{1}),\ldots ,(M_{n},d_{n})} 984:The metric space which most closely corresponds to 570:on WP for spaces which are not Hausdorff. The term 330:(1969) defines an isometry as an order-preserving 3091: 3061: 2986: 2956: 2891: 2871: 2851: 2825: 2799: 2777: 2725: 2705: 2679: 2644: 2556: 2536: 2361: 2045: 1992: 1940:{\displaystyle \scriptstyle N((4,1))={\sqrt {35}}} 1939: 1878: 1781: 1740: 1478: 1367: 1288:The following construction is useful to remember: 1164:One of the requirements of a metric space is that 3131:The introductory section ends with this passage: 2351: 2311: 2269: 2229: 2157: 2117: 2046:{\displaystyle \scriptstyle N((2,1))={\sqrt {7}}} 1993:{\displaystyle \scriptstyle N((2,2))={\sqrt {8}}} 1733: 1639: 1626: 1546: 1471: 1394: 1213:make sense? I'v never heard of such metric. :( -- 2778:{\displaystyle r={\frac {2s{\sqrt {3}}}{3}}: --> 2713:). The proper diameter is the length of a side 2571: 1015:...by using a different metric we can construct 975: 835:, which does not guarantee d(x,y)=d(y,x), or in 184:can be defined using a countable collections of 3546:A man-eating shark is a shark that eats humans. 149:should be merged. A topology is defined on the 3533:Pseudo-Riemannian manifold#Lorentzian manifold 1782:{\displaystyle \scriptstyle \langle .\rangle } 1148:Talk:Metric (mathematics)#Content syncronizing 358:I expect that Willard's and our definition at 3306:Metric space#Generalizations of metric spaces 2427:Me again. I would like to propose to change M 1283:I found the following material on this page: 581:Knowledge:WikiProject_Mathematics/Conventions 257:where the topology is defined by a family of 8: 2639: 2574: 1816: 1804: 1775: 1769: 122:different metrics can be used to define the 4007:possible terminology; it just needs to be 3173:Then change it lower in the text. Verdana 3076: 3031: 2971: 2926: 2884: 2864: 2838: 2812: 2792: 2756: 2747: 2739: 2718: 2692: 2657: 2625: 2600: 2586: 2569: 2549: 2529: 2375:in violation of the triangle inequality. 2350: 2349: 2310: 2309: 2300: 2287: 2268: 2267: 2228: 2227: 2218: 2205: 2185: 2175: 2165: 2156: 2155: 2116: 2115: 2106: 2093: 2081: 2035: 2005: 1982: 1952: 1929: 1899: 1828: 1802: 1766: 1752:and the induced topology agrees with the 1732: 1731: 1722: 1709: 1696: 1674: 1661: 1648: 1638: 1637: 1625: 1624: 1615: 1596: 1577: 1558: 1545: 1544: 1535: 1510: 1498: 1470: 1469: 1460: 1441: 1422: 1403: 1393: 1392: 1390: 1356: 1343: 1318: 1305: 1296: 201:My point is if your are talking about a 3520:"; "indefinite or mixed signature" in " 3482:Waiting 3 more days for your reaction. 3087: 3057: 2982: 2952: 1121:for the clarification -- sometimes I'm 3666:That's okay; nobody else does, either. 621:Homeomorphism and continuous functions 438:Knowledge talk:WikiProject Mathematics 334:(1-1), but says two metric spaces are 44:Do not edit the contents of this page. 3619:Yes, and you're living in a huge one. 7: 3474:And another question. No doubt that 756:d((x,y), (z,w)) = |x - z| + |y - w| 3976:It is not about proving you wrong, 3529:Metric tensor#Signature of a metric 976:'our intuitive', 'interesting', POV 1894:be the associated norm. Note that 24: 3518:Generalizations of the derivative 3516:; "indefinite metric tensor" in " 2067:are both just the usual norms on 1023:"Interesting" shows POV. It's a 742:british rail metric not a metric? 711:hope that answers your question. 3682:The Force is strong in this one! 3510:Topological quantum field theory 3152:real numbers as its distances. 3111:of points, namely zero distance. 2587: 797:Is the first condition really a 29: 3418:Pseudo-Euclidean space#Distance 3099:has nothing to do with x and y 2520:The picture given to represent 474:that is further required to be 3498:Indefinite inner product space 3462:Questions to User:Verdana Bold 3444:" is mentioned in "See also". 3048: 3036: 3019:18:03, 17 September 2014 (UTC) 2943: 2931: 2674: 2659: 2622: 2610: 2346: 2334: 2328: 2316: 2306: 2280: 2264: 2252: 2246: 2234: 2224: 2198: 2152: 2140: 2134: 2122: 2112: 2086: 2029: 2026: 2014: 2011: 1976: 1973: 1961: 1958: 1923: 1920: 1908: 1905: 1728: 1702: 1680: 1654: 1621: 1589: 1583: 1551: 1541: 1503: 1466: 1434: 1362: 1336: 1324: 1298: 684:. The smallest possible such 640:Right now the article reads: 613:04:34, 25 September 2005 (UTC) 601:18:22, 24 September 2005 (UTC) 588:00:29, 24 September 2005 (UTC) 549:22:36, 22 September 2005 (UTC) 525:07:38, 24 September 2005 (UTC) 515:16:47, 22 September 2005 (UTC) 487:18:06, 22 September 2005 (UTC) 448:16:32, 22 September 2005 (UTC) 436:I will raise this question at 424:17:00, 22 September 2005 (UTC) 411:16:39, 22 September 2005 (UTC) 395:08:58, 22 September 2005 (UTC) 378:15:47, 21 September 2005 (UTC) 311:08:55, 21 September 2005 (UTC) 1: 4059:of a (3,1) signature. Verdana 3662:But I do not understand you. 3639:Perhaps easier to understand: 2787:centered at the circumcenter 1486:is a metric space, where the 1273:21:55, 24 November 2008 (UTC) 1160:Non-commutative metric space? 1130:06:07, 5 September 2007 (UTC) 1101:08:52, 1 September 2007 (UTC) 1083:08:42, 1 September 2007 (UTC) 1055:06:44, 1 September 2007 (UTC) 1044:04:35, 1 September 2007 (UTC) 734:02:52, 30 December 2005 (UTC) 721:20:27, 29 December 2005 (UTC) 705:20:20, 29 December 2005 (UTC) 344:(onto) isometry between them. 188:(which should be merged with 2605: 2511:07:19, 30 October 2010 (UTC) 1257:22:08, 8 February 2008 (UTC) 1232:08:57, 7 November 2007 (UTC) 1218:07:32, 7 November 2007 (UTC) 1199:21:31, 11 October 2007 (UTC) 1185:21:27, 11 October 2007 (UTC) 1019:non-Euclidean geometries... 971:00:39, 30 October 2010 (UTC) 652:if there exists some number 631:10:40, 30 October 2005 (UTC) 3929:02:04, 28 June 2015 (UTC) 3821:pseudo-Riemannian manifolds 3794: 3504:"; "indefinite metric" in " 2489:16:05, 9 October 2010 (UTC) 2394:d_2(f(x), f(y)) ≥ d_1(x, y) 1155:17:52, 8 October 2007 (UTC) 816:d(A, A) ≤ d(A, B) + d(B, A) 470:be related by an isometric 349:Counterexamples in Topology 4154: 4070:but it sounds intriguing, 3850:pseudo-Riemannian manifold 3062:{\displaystyle d(x,y)=0\,} 2957:{\displaystyle d(x,y)=0\,} 2917:Identity of Indiscernibles 2471:space; see the article on 2454:23:28, 19 April 2009 (UTC) 2409:11:31, 17 April 2009 (UTC) 787:04:59, 10 March 2006 (UTC) 292:19:36, 11 March 2012 (UTC) 261:. So I think the split in 109:article has become to long 4136:08:49, 28 June 2015 (UTC) 4119:08:05, 28 June 2015 (UTC) 4107:02:22, 28 June 2015 (UTC) 4081:08:05, 28 June 2015 (UTC) 4066:08:05, 28 June 2015 (UTC) 4046:08:05, 28 June 2015 (UTC) 4031:08:05, 28 June 2015 (UTC) 3998:08:05, 28 June 2015 (UTC) 3972:08:05, 28 June 2015 (UTC) 3892:Okay, look, both of you: 3883:15:11, 28 June 2015 (UTC) 3867:Sorry, I was thinking of 3862:00:38, 28 June 2015 (UTC) 3843:00:19, 28 June 2015 (UTC) 3825:semi-Riemannian manifolds 3809:00:08, 28 June 2015 (UTC) 3790:17:41, 27 June 2015 (UTC) 3770:07:35, 26 June 2015 (UTC) 3745:06:02, 26 June 2015 (UTC) 3705:d² = x² + y² +z² + (-ct²) 3646:d² = x² + y² +z² + (-ct²) 3588:16:29, 25 June 2015 (UTC) 3572:distance can be negative) 3492:14:53, 25 June 2015 (UTC) 3454:17:11, 22 June 2015 (UTC) 3430:09:30, 22 June 2015 (UTC) 3335:17:56, 22 June 2015 (UTC) 3318:08:13, 22 June 2015 (UTC) 3273:20:08, 20 June 2015 (UTC) 3260:17:56, 22 June 2015 (UTC) 3237:The next sentence reads: 3233:17:56, 22 June 2015 (UTC) 3180:17:56, 22 June 2015 (UTC) 3121:20:16, 20 June 2015 (UTC) 3107:having one property as a 2912:02:18, 1 March 2013 (UTC) 2389:23:48, 10 July 2008 (UTC) 1004:popularization asserting 955:03:05, 25 July 2006 (UTC) 940:21:46, 24 July 2006 (UTC) 931:06:53, 22 July 2006 (UTC) 910:19:20, 21 July 2006 (UTC) 884:21:04, 18 July 2006 (UTC) 866:13:49, 18 July 2006 (UTC) 768:00:47, 9 March 2006 (UTC) 495:By the way, PlanetMath's 119:are separate articles too 3869:sub-Riemannian manifolds 3308:section, not in lead. -- 1789:be the inner product on 846:22:44, 3 July 2006 (UTC) 725:That's the problem with 203:topological vector space 180:should not be merged. A 137:12:28, 8 Apr 2005 (UTC) 3598:this one took 2½ hours 3278:That part was added by 2921:Why does the condition 2706:{\displaystyle n\geq 1} 1375:are metric spaces, and 1205:computer memory metric? 903:Talk:Norm (mathematics) 594:compact Hausdorff space 241:19:36, 2 May 2005 (UTC) 218:17:27, 8 Apr 2005 (UTC) 157:16:19, 2005 Apr 8 (UTC) 4126:Happy Nobel ceremony. 3632:d² = x² + y² +z² - ct² 3559:Hyphen#Varied_meanings 3476:pseudo-Euclidean space 3352:Pseudo-Euclidean space 3093: 3063: 2988: 2958: 2893: 2873: 2853: 2827: 2801: 2780: 2727: 2707: 2681: 2680:{\displaystyle (2n+1)} 2646: 2558: 2538: 2364: 2047: 1994: 1941: 1880: 1783: 1759: 1742: 1480: 1369: 1021: 990: 980:Current article says: 793:Superfluous condition? 750:||(x,y)|| = |x| + |y| 562:for the 1-1 case, and 3094: 3092:{\displaystyle x=y\,} 3064: 2989: 2987:{\displaystyle x=y\,} 2959: 2894: 2874: 2854: 2828: 2802: 2781: 2728: 2708: 2682: 2647: 2559: 2539: 2365: 2048: 1995: 1942: 1881: 1784: 1743: 1488:normed product metric 1481: 1370: 1285: 1227:. I've removed it. -- 1035:interesting, because 1013: 982: 347:Steen and Steenbach, 249:I just learned about 42:of past discussions. 3075: 3030: 2970: 2925: 2883: 2863: 2837: 2811: 2791: 2738: 2717: 2691: 2656: 2568: 2548: 2528: 2475:for further details. 2473:metrization theorems 2080: 2004: 1951: 1898: 1801: 1793:given by the formula 1765: 1497: 1389: 1295: 1190:Yes, they're called 1142:Content syncronizing 1031:that the geometries 799:definition condition 263:metric (mathematics) 96:metric (mathematics) 81:metric (mathematics) 3568:(but corrected: the 2852:{\displaystyle ABO} 2826:{\displaystyle ABC} 2381:skeptical scientist 560:isometric embedding 554:I prefer the names 178:normed vector space 147:normed vector space 117:normed vector space 3089: 3088: 3059: 3058: 2984: 2983: 2954: 2953: 2889: 2869: 2849: 2823: 2797: 2775: 2723: 2703: 2677: 2642: 2554: 2544:in a metric space 2534: 2359: 2043: 2042: 1990: 1989: 1937: 1936: 1876: 1864: 1779: 1778: 1738: 1476: 1365: 1279:Corrected an error 1245:mathematical model 1238:Hyperbolic example 1192:quasimetric spaces 1000:(BTW, I recall an 190:norm (mathematics) 174:Norm (mathematics) 143:norm (mathematics) 113:norm (mathematics) 3960: 3950: 3936:comment added by 3864: 3674:Yes yes; so what? 3562: 3554: 3552: 3541: 3022: 3005:comment added by 2892:{\displaystyle O} 2872:{\displaystyle C} 2800:{\displaystyle O} 2767: 2761: 2726:{\displaystyle s} 2628: 2608: 2557:{\displaystyle M} 2537:{\displaystyle A} 2190: 2180: 2170: 2040: 1987: 1934: 1225:original research 833:quasimetric space 636:Is this an error? 151:topological space 72: 71: 54: 53: 48:current talk page 18:Talk:Metric space 4145: 4115: 4077: 4062: 4042: 4027: 3994: 3968: 3954: 3949: 3930: 3925: 3875: 3847: 3835: 3817:reliable sources 3795:It's very simple 3741: 3574:fits better in " 3556: 3553: 3543: 3539: 3522:Metric signature 3442:Metric signature 3331: 3285: 3256: 3229: 3176: 3159:a metric space. 3098: 3096: 3095: 3090: 3068: 3066: 3065: 3060: 3021: 2999: 2993: 2991: 2990: 2985: 2963: 2961: 2960: 2955: 2898: 2896: 2895: 2890: 2878: 2876: 2875: 2870: 2859:is further from 2858: 2856: 2855: 2850: 2832: 2830: 2829: 2824: 2806: 2804: 2803: 2798: 2786: 2783: 2782: 2776: 2768: 2763: 2762: 2757: 2748: 2732: 2730: 2729: 2724: 2712: 2710: 2709: 2704: 2686: 2684: 2683: 2678: 2651: 2649: 2648: 2643: 2629: 2626: 2609: 2601: 2590: 2563: 2561: 2560: 2555: 2543: 2541: 2540: 2535: 2370: 2367: 2366: 2360: 2355: 2354: 2315: 2314: 2305: 2304: 2292: 2291: 2273: 2272: 2233: 2232: 2223: 2222: 2210: 2209: 2191: 2186: 2181: 2176: 2171: 2166: 2161: 2160: 2121: 2120: 2111: 2110: 2098: 2097: 2052: 2050: 2049: 2044: 2041: 2036: 1999: 1997: 1996: 1991: 1988: 1983: 1946: 1944: 1943: 1938: 1935: 1930: 1885: 1883: 1882: 1877: 1869: 1868: 1788: 1786: 1785: 1780: 1754:product topology 1747: 1745: 1744: 1739: 1737: 1736: 1727: 1726: 1714: 1713: 1701: 1700: 1679: 1678: 1666: 1665: 1653: 1652: 1643: 1642: 1630: 1629: 1620: 1619: 1601: 1600: 1582: 1581: 1563: 1562: 1550: 1549: 1540: 1539: 1515: 1514: 1485: 1483: 1482: 1477: 1475: 1474: 1465: 1464: 1446: 1445: 1427: 1426: 1408: 1407: 1398: 1397: 1374: 1372: 1371: 1366: 1361: 1360: 1348: 1347: 1323: 1322: 1310: 1309: 1098:Mark H Wilkinson 1052:Mark H Wilkinson 837:semimetric space 731:Plowboylifestyle 702:Plowboylifestyle 501:and MathWorld's 321:General Topology 319:Steven Willard, 68: 56: 55: 33: 32: 26: 4153: 4152: 4148: 4147: 4146: 4144: 4143: 4142: 4128:Boris Tsirelson 4113: 4109: 4075: 4060: 4040: 4025: 4012: 3992: 3966: 3931: 3923: 3915: 3873: 3833: 3797: 3782:Boris Tsirelson 3762:Boris Tsirelson 3739: 3593: 3580:Boris Tsirelson 3502:Minkowski space 3484:Boris Tsirelson 3464: 3446:Boris Tsirelson 3422:Boris Tsirelson 3348:Minkowski space 3329: 3296:generalizations 3294:There are some 3279: 3254: 3227: 3185:Which makes it 3174: 3129: 3073: 3072: 3028: 3027: 3000: 2968: 2967: 2923: 2922: 2919: 2881: 2880: 2861: 2860: 2835: 2834: 2833:, any point in 2809: 2808: 2789: 2788: 2749: 2735: 2734: 2715: 2714: 2689: 2688: 2654: 2653: 2566: 2565: 2546: 2545: 2526: 2525: 2518: 2481:Boris Tsirelson 2461: 2442: 2438: 2434: 2430: 2425: 2421: 2417: 2396: 2296: 2283: 2214: 2201: 2102: 2089: 2077: 2076: 2065: 2058: 2002: 2001: 1949: 1948: 1896: 1895: 1863: 1862: 1857: 1848: 1847: 1839: 1829: 1799: 1798: 1763: 1762: 1718: 1705: 1692: 1670: 1657: 1644: 1611: 1592: 1573: 1554: 1531: 1506: 1495: 1494: 1456: 1437: 1418: 1399: 1387: 1386: 1379:is any norm on 1352: 1339: 1314: 1301: 1293: 1292: 1281: 1240: 1207: 1162: 1144: 978: 947:Oleg Alexandrov 923:Oleg Alexandrov 824:0 ≤ 2 × d(A, B) 795: 744: 713:Oleg Alexandrov 644:A metric space 638: 623: 598:Septentrionalis 546:Oleg Alexandrov 300: 269:was reasonable. 102:. Reasons were 88: 64: 30: 22: 21: 20: 12: 11: 5: 4151: 4149: 4141: 4140: 4139: 4138: 4121: 4120: 4092: 4091: 4090: 4083: 4082: 4068: 4067: 4048: 4047: 4033: 4032: 4017: 4016: 4001: 4000: 3999: 3979: 3974: 3973: 3913: 3898: 3897: 3890: 3889: 3888: 3887: 3886: 3885: 3874:Sławomir Biały 3834:Sławomir Biały 3796: 3793: 3775: 3774: 3773: 3772: 3758: 3754: 3747: 3746: 3736: 3731: 3730: 3726: 3725: 3720: 3719: 3715: 3714: 3709: 3708: 3707: 3706: 3700: 3699: 3692: 3691: 3684: 3683: 3676: 3675: 3668: 3667: 3660: 3659: 3655: 3654: 3650: 3649: 3648: 3647: 3641: 3640: 3636: 3635: 3634: 3633: 3627: 3626: 3621: 3620: 3613: 3612: 3608: 3607: 3602: 3601: 3591: 3551: 3550: 3547: 3506:Suraj N. Gupta 3472: 3471: 3463: 3460: 3459: 3458: 3457: 3456: 3435: 3434: 3433: 3432: 3411: 3410: 3409: 3408: 3407: 3406: 3405: 3404: 3385: 3384: 3383: 3382: 3381: 3380: 3370: 3369: 3368: 3367: 3366: 3365: 3358: 3357: 3356: 3355: 3341: 3340: 3339: 3338: 3337: 3336: 3321: 3320: 3302: 3301: 3300: 3262: 3261: 3247: 3246: 3235: 3234: 3222: 3221: 3210: 3209: 3205: 3204: 3200: 3199: 3195: 3194: 3184: 3182: 3181: 3169: 3128: 3125: 3124: 3123: 3086: 3083: 3080: 3056: 3053: 3050: 3047: 3044: 3041: 3038: 3035: 2981: 2978: 2975: 2951: 2948: 2945: 2942: 2939: 2936: 2933: 2930: 2918: 2915: 2888: 2868: 2848: 2845: 2842: 2822: 2819: 2816: 2796: 2774: 2771: 2766: 2760: 2755: 2752: 2746: 2743: 2722: 2702: 2699: 2696: 2676: 2673: 2670: 2667: 2664: 2661: 2641: 2638: 2635: 2632: 2624: 2621: 2618: 2615: 2612: 2607: 2604: 2599: 2596: 2593: 2589: 2585: 2582: 2579: 2576: 2573: 2553: 2533: 2517: 2514: 2477: 2476: 2460: 2457: 2440: 2436: 2432: 2428: 2424: 2419: 2415: 2412: 2395: 2392: 2373: 2372: 2371: 2358: 2353: 2348: 2345: 2342: 2339: 2336: 2333: 2330: 2327: 2324: 2321: 2318: 2313: 2308: 2303: 2299: 2295: 2290: 2286: 2282: 2279: 2276: 2271: 2266: 2263: 2260: 2257: 2254: 2251: 2248: 2245: 2242: 2239: 2236: 2231: 2226: 2221: 2217: 2213: 2208: 2204: 2200: 2197: 2194: 2189: 2184: 2179: 2174: 2169: 2164: 2159: 2154: 2151: 2148: 2145: 2142: 2139: 2136: 2133: 2130: 2127: 2124: 2119: 2114: 2109: 2105: 2101: 2096: 2092: 2088: 2085: 2072: 2063: 2056: 2039: 2034: 2031: 2028: 2025: 2022: 2019: 2016: 2013: 2010: 1986: 1981: 1978: 1975: 1972: 1969: 1966: 1963: 1960: 1957: 1933: 1928: 1925: 1922: 1919: 1916: 1913: 1910: 1907: 1904: 1888: 1887: 1886: 1875: 1872: 1867: 1861: 1858: 1856: 1853: 1850: 1849: 1846: 1843: 1840: 1838: 1835: 1834: 1832: 1827: 1824: 1821: 1818: 1815: 1812: 1809: 1806: 1794: 1777: 1774: 1771: 1758: 1750: 1749: 1735: 1730: 1725: 1721: 1717: 1712: 1708: 1704: 1699: 1695: 1691: 1688: 1685: 1682: 1677: 1673: 1669: 1664: 1660: 1656: 1651: 1647: 1641: 1636: 1633: 1628: 1623: 1618: 1614: 1610: 1607: 1604: 1599: 1595: 1591: 1588: 1585: 1580: 1576: 1572: 1569: 1566: 1561: 1557: 1553: 1548: 1543: 1538: 1534: 1530: 1527: 1524: 1521: 1518: 1513: 1509: 1505: 1502: 1490:is defined by 1473: 1468: 1463: 1459: 1455: 1452: 1449: 1444: 1440: 1436: 1433: 1430: 1425: 1421: 1417: 1414: 1411: 1406: 1402: 1396: 1364: 1359: 1355: 1351: 1346: 1342: 1338: 1335: 1332: 1329: 1326: 1321: 1317: 1313: 1308: 1304: 1300: 1286: 1280: 1277: 1276: 1275: 1239: 1236: 1235: 1234: 1206: 1203: 1202: 1201: 1161: 1158: 1143: 1140: 1139: 1138: 1137: 1136: 1135: 1134: 1133: 1132: 1123:too suspicious 1108: 1107: 1106: 1105: 1104: 1103: 1088: 1087: 1086: 1085: 1072: 1071: 1070: 1069: 1058: 1057: 977: 974: 961:definition. -- 958: 957: 919: 918: 917: 916: 915: 914: 913: 912: 891: 890: 889: 888: 887: 886: 871: 870: 869: 868: 857:rather than a 840: 826: 825: 818: 817: 794: 791: 790: 789: 759:Any thoughts? 743: 740: 739: 738: 737: 736: 698: 697: 688:is called the 637: 634: 622: 619: 618: 617: 616: 615: 603: 530: 529: 528: 527: 492: 491: 490: 489: 478:(and hence an 451: 450: 433: 432: 431: 430: 429: 428: 427: 426: 381: 380: 355: 354: 353: 352: 345: 338:if there is a 324: 299: 296: 295: 294: 275: 273: 272: 271: 270: 255:uniform spaces 244: 243: 226: 222: 221: 220: 219: 209: 208: 207: 206: 196: 195: 194: 193: 168: 167: 166: 165: 159: 158: 132: 131: 120: 110: 87: 73: 70: 69: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 4150: 4137: 4133: 4129: 4125: 4124: 4123: 4122: 4118: 4111: 4110: 4108: 4104: 4100: 4096: 4088: 4087: 4086: 4080: 4073: 4072: 4071: 4065: 4058: 4053: 4052: 4051: 4045: 4038: 4037: 4036: 4030: 4023: 4019: 4018: 4014: 4013: 4010: 4006: 3997: 3990: 3986: 3982: 3981: 3980: 3977: 3971: 3964: 3963: 3962: 3959:your name :-) 3958: 3951: 3947: 3943: 3939: 3935: 3928: 3919: 3912: 3909: 3906: 3902: 3895: 3894: 3893: 3884: 3880: 3876: 3870: 3866: 3865: 3863: 3859: 3855: 3851: 3846: 3845: 3844: 3840: 3836: 3830: 3829:metric tensor 3826: 3822: 3818: 3813: 3812: 3811: 3810: 3806: 3802: 3792: 3791: 3787: 3783: 3780: 3771: 3767: 3763: 3759: 3755: 3751: 3750: 3749: 3748: 3744: 3737: 3733: 3732: 3728: 3727: 3722: 3721: 3717: 3716: 3711: 3710: 3704: 3703: 3702: 3701: 3697: 3696: 3695: 3689: 3688: 3687: 3681: 3680: 3679: 3673: 3672: 3671: 3665: 3664: 3663: 3657: 3656: 3652: 3651: 3645: 3644: 3643: 3642: 3638: 3637: 3631: 3630: 3629: 3628: 3623: 3622: 3618: 3617: 3616: 3610: 3609: 3604: 3603: 3599: 3595: 3594: 3590: 3589: 3585: 3581: 3577: 3573: 3569: 3564: 3560: 3548: 3545: 3544: 3542: 3536: 3534: 3530: 3525: 3523: 3519: 3515: 3511: 3507: 3503: 3499: 3494: 3493: 3489: 3485: 3480: 3477: 3469: 3468: 3467: 3461: 3455: 3451: 3447: 3443: 3439: 3438: 3437: 3436: 3431: 3427: 3423: 3419: 3415: 3414: 3413: 3412: 3402: 3398: 3393: 3392: 3391: 3390: 3389: 3388: 3387: 3386: 3376: 3375: 3374: 3373: 3372: 3371: 3364: 3363: 3362: 3361: 3360: 3359: 3353: 3349: 3345: 3344: 3343: 3342: 3334: 3327: 3326: 3325: 3324: 3323: 3322: 3319: 3315: 3311: 3307: 3303: 3299: 3297: 3292: 3291: 3289: 3283: 3277: 3276: 3275: 3274: 3270: 3266: 3259: 3252: 3251: 3250: 3244: 3240: 3239: 3238: 3232: 3224: 3223: 3220: 3219:intransitive. 3216: 3212: 3211: 3207: 3206: 3202: 3201: 3197: 3196: 3192: 3191: 3190: 3188: 3179: 3172: 3171: 3170: 3167: 3165: 3160: 3158: 3153: 3151: 3147: 3143: 3139: 3137: 3132: 3126: 3122: 3118: 3114: 3110: 3106: 3102: 3084: 3081: 3078: 3071: 3054: 3051: 3045: 3042: 3039: 3033: 3025: 3024: 3023: 3020: 3016: 3012: 3008: 3004: 2996: 2979: 2976: 2973: 2966: 2949: 2946: 2940: 2937: 2934: 2928: 2916: 2914: 2913: 2909: 2905: 2900: 2886: 2866: 2846: 2843: 2840: 2820: 2817: 2814: 2794: 2772: 2769: 2764: 2758: 2753: 2750: 2744: 2741: 2720: 2700: 2697: 2694: 2671: 2668: 2665: 2662: 2636: 2633: 2630: 2619: 2616: 2613: 2602: 2597: 2594: 2591: 2583: 2580: 2577: 2551: 2531: 2523: 2515: 2513: 2512: 2508: 2504: 2500: 2496: 2493:My apologies 2491: 2490: 2486: 2482: 2474: 2470: 2466: 2465: 2464: 2459:Metrizability 2458: 2456: 2455: 2450: 2446: 2413: 2411: 2410: 2406: 2402: 2393: 2391: 2390: 2386: 2382: 2376: 2356: 2343: 2340: 2337: 2331: 2325: 2322: 2319: 2301: 2297: 2293: 2288: 2284: 2277: 2274: 2261: 2258: 2255: 2249: 2243: 2240: 2237: 2219: 2215: 2211: 2206: 2202: 2195: 2192: 2187: 2182: 2177: 2172: 2167: 2162: 2149: 2146: 2143: 2137: 2131: 2128: 2125: 2107: 2103: 2099: 2094: 2090: 2083: 2075: 2074: 2073: 2070: 2066: 2059: 2053:, so that if 2037: 2032: 2023: 2020: 2017: 2008: 1984: 1979: 1970: 1967: 1964: 1955: 1931: 1926: 1917: 1914: 1911: 1902: 1893: 1873: 1870: 1865: 1859: 1854: 1851: 1844: 1841: 1836: 1830: 1825: 1822: 1819: 1813: 1810: 1807: 1797: 1796: 1795: 1792: 1772: 1757: 1755: 1723: 1719: 1715: 1710: 1706: 1697: 1693: 1689: 1686: 1683: 1675: 1671: 1667: 1662: 1658: 1649: 1645: 1634: 1631: 1616: 1612: 1608: 1605: 1602: 1597: 1593: 1586: 1578: 1574: 1570: 1567: 1564: 1559: 1555: 1536: 1532: 1528: 1525: 1522: 1519: 1516: 1511: 1507: 1500: 1493: 1492: 1491: 1489: 1461: 1457: 1453: 1450: 1447: 1442: 1438: 1431: 1428: 1423: 1419: 1415: 1412: 1409: 1404: 1400: 1384: 1382: 1378: 1357: 1353: 1349: 1344: 1340: 1333: 1330: 1327: 1319: 1315: 1311: 1306: 1302: 1289: 1284: 1278: 1274: 1270: 1266: 1261: 1260: 1259: 1258: 1254: 1250: 1246: 1237: 1233: 1230: 1226: 1222: 1221: 1220: 1219: 1216: 1212: 1204: 1200: 1197: 1193: 1189: 1188: 1187: 1186: 1183: 1179: 1175: 1171: 1167: 1159: 1157: 1156: 1153: 1149: 1141: 1131: 1128: 1124: 1120: 1116: 1115: 1114: 1113: 1112: 1111: 1110: 1109: 1102: 1099: 1094: 1093: 1092: 1091: 1090: 1089: 1084: 1081: 1076: 1075: 1074: 1073: 1066: 1062: 1061: 1060: 1059: 1056: 1053: 1048: 1047: 1046: 1045: 1042: 1038: 1034: 1030: 1026: 1020: 1018: 1012: 1009: 1007: 1003: 998: 995: 989: 987: 986:our intuitive 981: 973: 972: 968: 964: 956: 952: 948: 944: 943: 942: 941: 938: 933: 932: 928: 924: 911: 908: 904: 899: 898: 897: 896: 895: 894: 893: 892: 885: 882: 877: 876: 875: 874: 873: 872: 867: 864: 860: 856: 852: 851: 850: 849: 848: 847: 844: 838: 834: 829: 823: 822: 821: 815: 814: 813: 811: 806: 804: 800: 792: 788: 785: 781: 777: 773: 772: 771: 770: 769: 766: 760: 757: 754: 751: 748: 741: 735: 732: 728: 724: 723: 722: 718: 714: 709: 708: 707: 706: 703: 695: 691: 687: 683: 679: 675: 671: 667: 663: 659: 655: 651: 647: 643: 642: 641: 635: 633: 632: 629: 620: 614: 611: 608: 604: 602: 599: 595: 591: 590: 589: 586: 582: 577: 573: 572:quasi-compact 569: 565: 561: 557: 556:isometric map 553: 552: 551: 550: 547: 543: 539: 535: 526: 523: 518: 517: 516: 513: 510: 506: 505: 500: 499: 494: 493: 488: 485: 481: 477: 473: 469: 465: 460: 455: 454: 453: 452: 449: 446: 443: 439: 435: 434: 425: 422: 418: 414: 413: 412: 409: 406: 402: 398: 397: 396: 393: 389: 385: 384: 383: 382: 379: 376: 373: 369: 365: 361: 357: 356: 350: 346: 343: 342: 337: 333: 329: 326:Eduard Čhec, 325: 322: 318: 317: 315: 314: 313: 312: 309: 305: 297: 293: 289: 285: 282: 278: 277: 276: 268: 264: 260: 259:pseudometrics 256: 252: 248: 247: 246: 245: 242: 239: 235: 231: 227: 224: 223: 217: 213: 212: 211: 210: 204: 200: 199: 198: 197: 191: 187: 183: 182:Frechet space 179: 175: 172: 171: 170: 169: 163: 162: 161: 160: 156: 152: 148: 144: 140: 139: 138: 136: 129: 125: 121: 118: 114: 111: 108: 105: 104: 103: 101: 97: 93: 86: 82: 78: 74: 67: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 4116: 4084: 4078: 4069: 4063: 4056: 4049: 4043: 4034: 4028: 4021: 4008: 4004: 3995: 3989:immediately. 3988: 3984: 3978: 3975: 3969: 3956: 3952: 3938:Verdana Bold 3932:— Preceding 3926: 3920: 3916: 3910: 3907: 3903: 3899: 3891: 3823:rather than 3798: 3776: 3742: 3724:themselves). 3693: 3685: 3677: 3669: 3661: 3614: 3597: 3592:___________ 3571: 3567: 3565: 3555: 3537: 3526: 3495: 3481: 3473: 3465: 3400: 3396: 3350:, a case of 3332: 3293: 3282:Verdana Bold 3263: 3257: 3248: 3242: 3236: 3230: 3218: 3186: 3183: 3177: 3168: 3163: 3161: 3156: 3154: 3149: 3146:metric space 3145: 3141: 3140: 3135: 3133: 3130: 3108: 3104: 3100: 3001:— Preceding 2997: 2920: 2901: 2519: 2503:Lovy Singhal 2498: 2492: 2478: 2468: 2462: 2426: 2397: 2377: 2374: 2068: 2061: 2054: 1891: 1889: 1790: 1760: 1751: 1487: 1385: 1380: 1376: 1290: 1287: 1282: 1241: 1208: 1177: 1173: 1169: 1165: 1163: 1145: 1122: 1118: 1064: 1036: 1032: 1028: 1024: 1022: 1016: 1014: 1010: 1002:Irving Adler 999: 993: 991: 985: 983: 979: 959: 934: 920: 858: 854: 830: 827: 819: 809: 807: 798: 796: 762: 761: 758: 755: 752: 749: 745: 699: 693: 689: 685: 681: 677: 673: 669: 665: 661: 657: 656:, such that 653: 649: 645: 639: 624: 593: 575: 571: 567: 563: 559: 555: 531: 503: 497: 479: 475: 471: 467: 463: 458: 363: 348: 339: 335: 327: 320: 301: 274: 267:metric space 133: 127: 123: 107:metric space 100:metric space 92:metric space 89: 85:metric space 77:metric space 65: 43: 37: 3953:Hi Verdana 3611:Now then... 3514:Four-vector 3150:nonnegative 2904:kielejocain 2501:arise ..." 2495:User:Tsirel 2463:The phrase 2453:_X,_Y": --> 1146:Please see 1017:interesting 907:MisterSheik 765:82.42.16.20 607:Paul August 509:Paul August 442:Paul August 405:Paul August 372:Paul August 251:gage spaces 36:This is an 4095:WP:RD/Math 4057:definition 2785:s}" /: --> 2469:metrizable 1182:Loadmaster 881:Madmath789 810:a property 648:is called 544:standard! 538:convention 341:surjective 328:Point Sets 216:MathMartin 135:MathMartin 4099:Trovatore 3991:--Verdana 3854:Trovatore 3801:Trovatore 3777:See also 3713:distance. 3625:Einstein: 3527:See also 3288:this edit 3215:null cone 3148:has only 2627:for some 1211:this edit 963:Reineke80 937:Hammerite 855:condition 727:Knowledge 476:bijective 421:Trovatore 364:isometric 336:isometric 332:injection 302:Isn't an 186:semi-norm 153:page. -- 66:Archive 2 60:Archive 1 4009:standard 3955:if that 3946:contribs 3934:unsigned 3570:squared 3105:together 3015:contribs 3007:Catrincm 3003:unsigned 2737:s}": --> 2522:diameter 2516:Diameter 1117:OK, and 1068:neutral. 1063:So this 1037:interest 994:a priori 863:HellFire 859:property 690:diameter 672:for all 564:isometry 504:isometry 498:Isometry 480:isometry 417:isometry 401:Isometry 388:Isometry 368:Isometry 360:Isometry 304:isometry 298:Isometry 284:PJTraill 128:sameness 90:I split 3922:Verdana 3914:------ 3738:Verdana 3500:" and " 3226:Verdana 1383:, then 1265:Rgdboer 1249:Rgdboer 1229:Zundark 1196:Zundark 1029:opinion 650:bounded 576:compact 568:compact 534:webster 155:Fropuff 39:archive 4022:to me; 3512:, and 3310:CiaPan 3142:Wrong. 2687:-gon, 2422:-: --> 2071:, then 2000:, and 1215:CiaPan 1119:thanks 1065:byword 843:CiaPan 803:metric 801:for a 522:JahJah 468:spaces 392:JahJah 308:JahJah 75:Split 4097:. -- 3985:hate 3399:and - 3162:This 2899:is). 2879:than 2770:: --> 2439:and M 2431:and M 2173:: --> 1209:Does 1152:Míkka 1011:And: 585:Gauge 484:KSmrq 482:). -- 94:into 79:into 16:< 4132:talk 4117:Bøld 4103:talk 4079:Bøld 4064:Bøld 4044:Bøld 4029:Bøld 4005:best 3996:Bøld 3970:Bøld 3942:talk 3927:Bøld 3879:talk 3858:talk 3839:talk 3805:talk 3786:talk 3766:talk 3743:Bøld 3735:BJs. 3584:talk 3557:see 3531:and 3488:talk 3450:talk 3426:talk 3416:See 3333:Bøld 3314:talk 3269:talk 3265:Daqu 3258:Bøld 3231:Bøld 3187:very 3178:Bøld 3164:also 3117:talk 3113:Daqu 3109:pair 3101:each 3011:talk 2908:talk 2507:talk 2485:talk 2449:talk 2445:Taku 2423:X, Y 2405:talk 2401:Taku 2385:talk 2060:and 1269:talk 1253:talk 1194:. -- 1172:) = 1150:. `' 1025:fact 1006:Kant 967:talk 951:talk 927:talk 784:Talk 776:SNCF 717:talk 676:and 668:) ≤ 628:Tomo 583:. - 558:and 464:must 459:some 390:. -- 288:talk 265:and 238:Talk 176:and 145:and 124:same 115:and 98:and 83:and 3871:. 3508:", 3286:in 3157:not 3070:iff 2965:iff 2572:inf 2499:can 2418:, M 1291:If 1178:y,x 1170:x,y 1033:are 780:MFH 692:of 680:in 542:ISO 472:map 234:MFH 4134:) 4105:) 3961:. 3957:is 3948:) 3944:• 3881:) 3860:) 3841:) 3807:) 3788:) 3768:) 3586:) 3561:). 3490:) 3452:) 3428:) 3420:. 3316:) 3271:) 3144:A 3138:" 3119:) 3017:) 3013:• 2910:) 2779:s} 2698:≥ 2634:∈ 2606:¯ 2598:⊂ 2584:∈ 2509:) 2487:) 2451:) 2407:) 2387:) 2168:35 1947:, 1932:35 1852:− 1842:− 1826:⋅ 1817:⟩ 1805:⟨ 1776:⟩ 1770:⟨ 1756:. 1687:… 1606:… 1568:… 1451:… 1416:× 1413:… 1410:× 1331:… 1271:) 1255:) 1127:AC 1096:-- 1080:AC 1050:-- 1041:AC 969:) 953:) 929:) 841:-- 805:? 719:) 696:.' 520:-- 440:. 403:. 370:. 290:) 253:, 236:: 232:— 4130:( 4114:♥ 4101:( 4076:♥ 4061:♥ 4041:♥ 4026:♥ 3993:♥ 3967:♥ 3940:( 3924:♥ 3877:( 3856:( 3837:( 3803:( 3784:( 3764:( 3740:♥ 3582:( 3486:( 3448:( 3424:( 3401:i 3397:i 3395:+ 3330:♥ 3312:( 3284:: 3280:@ 3267:( 3255:♥ 3245:" 3241:" 3228:♥ 3175:♥ 3134:" 3115:( 3085:y 3082:= 3079:x 3055:0 3052:= 3049:) 3046:y 3043:, 3040:x 3037:( 3034:d 3009:( 2980:y 2977:= 2974:x 2950:0 2947:= 2944:) 2941:y 2938:, 2935:x 2932:( 2929:d 2906:( 2887:O 2867:C 2847:O 2844:B 2841:A 2821:C 2818:B 2815:A 2795:O 2773:s 2765:3 2759:3 2754:s 2751:2 2745:= 2742:r 2721:s 2701:1 2695:n 2675:) 2672:1 2669:+ 2666:n 2663:2 2660:( 2640:} 2637:A 2631:x 2623:) 2620:r 2617:, 2614:x 2611:( 2603:B 2595:A 2592:: 2588:R 2581:r 2578:2 2575:{ 2552:M 2532:A 2505:( 2483:( 2447:( 2441:2 2437:1 2433:2 2429:1 2420:2 2416:1 2414:M 2403:( 2383:( 2357:, 2352:) 2347:) 2344:1 2341:, 2338:4 2335:( 2332:, 2329:) 2326:2 2323:, 2320:2 2317:( 2312:( 2307:) 2302:2 2298:d 2294:, 2289:1 2285:d 2281:( 2278:N 2275:+ 2270:) 2265:) 2262:2 2259:, 2256:2 2253:( 2250:, 2247:) 2244:0 2241:, 2238:0 2235:( 2230:( 2225:) 2220:2 2216:d 2212:, 2207:1 2203:d 2199:( 2196:N 2193:= 2188:7 2183:+ 2178:8 2163:= 2158:) 2153:) 2150:1 2147:, 2144:4 2141:( 2138:, 2135:) 2132:0 2129:, 2126:0 2123:( 2118:( 2113:) 2108:2 2104:d 2100:, 2095:1 2091:d 2087:( 2084:N 2069:R 2064:2 2062:d 2057:1 2055:d 2038:7 2033:= 2030:) 2027:) 2024:1 2021:, 2018:2 2015:( 2012:( 2009:N 1985:8 1980:= 1977:) 1974:) 1971:2 1968:, 1965:2 1962:( 1959:( 1956:N 1927:= 1924:) 1921:) 1918:1 1915:, 1912:4 1909:( 1906:( 1903:N 1892:N 1874:. 1871:y 1866:] 1860:3 1855:2 1845:2 1837:3 1831:[ 1823:x 1820:= 1814:y 1811:, 1808:x 1791:R 1773:. 1748:, 1734:) 1729:) 1724:n 1720:y 1716:, 1711:n 1707:x 1703:( 1698:n 1694:d 1690:, 1684:, 1681:) 1676:1 1672:y 1668:, 1663:1 1659:x 1655:( 1650:1 1646:d 1640:( 1635:N 1632:= 1627:) 1622:) 1617:n 1613:y 1609:, 1603:, 1598:1 1594:y 1590:( 1587:, 1584:) 1579:n 1575:x 1571:, 1565:, 1560:1 1556:x 1552:( 1547:( 1542:) 1537:n 1533:d 1529:, 1526:. 1523:. 1520:. 1517:, 1512:1 1508:d 1504:( 1501:N 1472:) 1467:) 1462:n 1458:d 1454:, 1448:, 1443:1 1439:d 1435:( 1432:N 1429:, 1424:n 1420:M 1405:1 1401:M 1395:( 1381:R 1377:N 1363:) 1358:n 1354:d 1350:, 1345:n 1341:M 1337:( 1334:, 1328:, 1325:) 1320:1 1316:d 1312:, 1307:1 1303:M 1299:( 1267:( 1251:( 1176:( 1174:d 1168:( 1166:d 965:( 949:( 925:( 782:: 715:( 694:M 686:r 682:M 678:y 674:x 670:r 666:y 664:, 662:x 660:( 658:d 654:r 646:M 610:☎ 512:☎ 445:☎ 408:☎ 375:☎ 286:( 130:) 50:.

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