Definition talk:Metric
The notation $\{M,d\}$ for a metric space is not good. Rather, a metric space should be a ordered pair $(M,d)$.
The point is that, with the latter definition, it is clear from the object which of the two is the metric space, and which one is the metric. With the former definition, you would probably have to use the axiom of foundation, or something similar.
That may seem like an arcane point, but it is why the standard definition in all books that I am aware of uses an ordered pair. It is also conceptually the right definition. --lasserempe
I don't have enough experience with metric spaces to really say, but we might want to allow both, simply because of the number of pages that will reference it. And since we say at the top of the page that $\{M,d\}$ is a metric space, people shouldn't be overly confused. --Cynic (talk) 19:08, 7 March 2009 (UTC)
I'm using W. A. Sutherland's "Introduction to Metric and Topological Spaces" which uses the $\left\{{M, d}\right\}$ notation.
I'd take issue with "not good". Ultimately, nor is $\left({M, d}\right)$ because it still relies on convention that the metric follows the set it's on in the order that they're stated. Also, $\left({M, d}\right)$ adds a layer of complexity on the contributing elements, when in fact all that's needed are the set $M$ and the metric $d$, and the order that they appear is irrelevant except for the convention that the set happens to be stated first. $\left\{{d, M}\right\}$ means exactly the same thing as $\left\{{M, d}\right\}$ (one's a set, one's a mapping on that set, who cares what their order is?) whereas $\left({M, d}\right)$ adds an irrelevant level of structure on it that adds nothing to the content of the object except make it $\left\{{\left\{{M}\right\}, \left\{{M, d}\right\}}\right\}$ which is so "what's the point?" --Matt Westwood 21:55, 7 March 2009 (UTC)
The point is that you want the object you refer to as "the metric space" to determine both the base space and the metric. A set is an unordered object. Hence, if I tell you "the set $\{A,B\}$ is a metric space", how do you know which is the set, and which is the metric?
You may say, we look which of the two is a function defined on the Cartesian product of the other with itself - but how do you prove that it can't be the case for both? It would seem that you need the axiom of foundation for this. Otherwise, there may be the possibility that $A$ is a metric on $B$, but also $B$ is a metric on $A$, so the same set represents two potentially very different metric spaces!
Also, conceptually, there is a difference. A set takes a number of objects and combines them to a whole. They do not come in any particular order, or with any particular function: we think of all elements of a set to be on an equivalent level. (Of course this may change when the set has an additional structure.) That's not really what we want with a metric space (or a group, or a field, or a ring, a graph etc.): the different parts that make up the object have very different functions. This is why they are usually denoted as ordered pairs (or, more generally, tuples).
Of course the fault here lies mainly with Sutherland for using this notation in his book. I will stick with my claim that something that forces us to implicitly use the axiom of foundation every time we say anything about a metric space isn't particularly good notation!
Of course you can look at this as mainly an arcane argument. And I will agree that there is plenty of notation in mathematics that is not 100 percent formally correct, but everyone knows what is meant. And - perhaps unfortunately - it is virtually impossible to change such conventions, and we have to live with them. However, I am fairly confident in my assessment that the majority of textbooks, all lectures I have ever taken or taught, and also online sources such as Wikipedia use the ordered pair notation. As I have argued above, the "set" notation is inferior in many respects, and as it is a very basic concept that will come up again and again, I think it's worth having a discussion about this.
Now, I appreciate the point that there are already many pages that use this notation. My suggestion would be to change the definition to an ordered pair, and add a note that the other notation is also sometimes used. This should then, however, include a remark that this should be seen purely as notation; i.e. we do not really think of the metric space as a two-element set, but rather as a set together with a metric, where it is clear which is which. It isn't completely clear to me how to best formulate this, though. I don't expect you to put effort into changing it if you think it doesn't matter; I'm happy to make changes myself when I have the time. I just wanted to have a discussion rather than go around making such fundamental changes. -- lasserempe 22:49, 7 March 2009 (UTC)
I think I'm basically convinced. I wonder whether to take issue with the facility which delivered the course on topology that I took which used Sutherland as its basic text for not pointing this out. ;-) I'll change the definition and we can go and change the pages at leisure (I think I linked everywhere that uses the concept back to this page so that should be straightforward). One thing that puzzles me:
- there may be the possibility that $A$ is a metric on $B$, but also $B$ is a metric on $A$, so the same set represents two potentially very different metric spaces!
... how? --Matt Westwood 23:04, 7 March 2009 (UTC)
Further thought: is there any conceptual reason (beyond the fact that notationally it doesn't remove the possibility of there being more than 2 arguments) for $d \left({x, y}\right)$ not to be written and interpreted $d \left\{{x, y}\right\}$? --Matt Westwood 23:13, 7 March 2009 (UTC)
...and does the same apply to a Definition:Topological Space?
Hi,
regarding your first question, I haven't worked out the exact conditions of what one would need to get such a situation, but I'm pretty sure it's consistent with the usual axioms of set theory without foundation. (Foundation is what excludes the possibility e.g. for a set to contain itself as an element.) On the other hand, if we use a set theory where functions are not sets, then this problem disappears - so in a way it's an artificial result of the decision to base mathematics on sets alone ...
I see no concepual reason not to view as a metric (or any symmetric function of two arguments) as a function defined on the set of one- and two-element sets. I guess - apart from tradition - one reason not to use this as a definition is that whenever we write down a metric, we need to say that it is well-defined (i.e., doesn't depend on the order of elements), whereas with the usual definition, we don't need to do anything to define the function. Of course, ultimately we will need to make the same argument in order to show that it's symmetric, so this really is a bit of an arcane discussion ... (I like though that one of the conditions could be formulated as "d(A)=0 iff A is a one-element set ..." :P
Yes, my argument also applies to topological spaces, and to all other objects that are given by a set together with some additional structure. Thanks -- lasserempe 23:49, 7 March 2009 (UTC)
Right, I think that's the lot. --Matt Westwood 12:51, 8 March 2009 (UTC)
