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Let {X,T} be a topology, T the set of open subsets of X.


Definition: Three points x, y, z of X are in relation N (Nxyz, read "x is nearer to y than to z") iff

  1. there is a basis B of T and b in B such that x and y are in b but z is not and
  2. there is no basis C of T and c in C such that x and z are in c but not y.

For some topologies there are no points x, y, z in relation N, for example if T = {Ø,X} or T = P(X), but for others there are (e.g. for ones induced by a metric [my claim]).


Definition: A topology has property M1 iff

(x)(y) ((z) (z ≠ x & z ≠ y) → Nxyz) → x = y

(This is an analogue of d(xy) = 0 → x = y, the best one I can imagine).


Definition: A topology has property M2 iff

(x)(y)(z) Nxyz & Nyzx → Nzyx

(This is a kind of an analogue of d(xy) = d(yx), the best one I can imagine)


First (bunch of) question(s):

  1. Properties M1 and M2 do not capture the whole of the corresponding conditions of a metric. Can anyone figure out "better" definitions (e.g. an analogon of x = y → d(xy) = 0)?

  2. Can anyone figure out a property M3 that is an analogue of the triangle equality?

If it can be shown that no such property M3 is definable, the following becomes obsolete.

If such a definition can be made, we define:


Definition: A topology has property M (read "induces a metric") iff it has properties M1, M2, M3.


Second question:

Which topologies have property M, i.e. induce a metric? Are these "accidentally" exactly those that are induced by a metric?

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I don't have an answer to your questions, but, more conventionally, there is a standard theorem about which topological spaces admit a metric. It's beautiful, though not too easy -- it's one of the key points of a Point-Set topology course. Look in Munkres's Topology, or here: en.wikipedia.org/wiki/Metrizable_space. Also, if you like these kinds of questions, I highly recommend you read either Bourbaki's "General Topology" or Kelley's "General Topology". Both have a few generalizations of metrics that you might enjoy (in particular, uniformities). –  Ilya Grigoriev Dec 20 '09 at 0:39
    
Usage comments: "analogon" is much less common than the synonomous "analogue". "Has property X" or "is in relation N" are poor choices. Better is "Given a topology T on X, say that x is closer to y than to z if....". –  Theo Johnson-Freyd Dec 20 '09 at 3:16
    
You might just want to learn en.wikipedia.org/wiki/… . –  Qiaochu Yuan Dec 20 '09 at 4:46
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All of the important information of a metric can be recovered by its induced uniform space. This carries both the topological information as well as the uniform information (uniform convergence, completeness, et cetera). I don't believe you can get all of the information you want with just a topology. I am sure though, that one can only recover a metric up to topological equivalence. –  Harry Gindi Dec 20 '09 at 4:59
    
@Theo: I wanted to define the relation without bias. The proposed reading should be only a hint where the definition is aimed at. –  Hans Stricker Dec 20 '09 at 16:04
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2 Answers 2

up vote 11 down vote accepted

Your condition 1 is satisfied for all triples $x,y,z\in X$ such that $z\not\in\{x,y\}$ if the space is $T_1$.

Maybe reading a bit about uniform spaces and the corresponding metrizability results will be of help.

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Picking up Aaron's answer: Might there be a way to sensibly restrict the class of "allowed" (= "sensible") bases. What is T<sub>1</sub>? –  Hans Stricker Dec 20 '09 at 1:09
    
There is no more reference to bases anymore in your (edited) answer. My follow-up question remains (sorry): what space is T_1= –  Hans Stricker Dec 20 '09 at 1:13
    
I found the term T_1 linked and will follow this. Thanks! –  Hans Stricker Dec 20 '09 at 1:34
    
In particular, HP Stricker's claim is patently false. –  Theo Johnson-Freyd Dec 20 '09 at 3:19
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+1 for mentioning uniform spaces –  Harry Gindi Dec 20 '09 at 8:53
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I'm not sure your "nearness" relation is the right definition to make. It makes sense on the face of it, but for example I think you can generate the usual topology on the plane by tubular neighborhoods of half-circles. For three points x,y,z on a line in that order, I don't think you get any relationship. Using the usual disk basis you'd satisfy condition (1) for Nxyz, but you could violate condition (2) using the other basis I just suggested, if the tubular neighborhood is sufficiently small.

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However, I too have wondered which topologies can be induced by metrics. I think the first step in this direction would be to look for a topology which is <i>not</i> induced by a metric... –  Aaron Mazel-Gee Dec 20 '09 at 0:25
    
I thank you for this. Maybe the look for "unusual" bases should be sharpened (for beginners like me). Coming the standard way I thought only of disk bases. But maybe my line of thought can be "rescued"? If not: maybe it helps understanding the difference between topologies and metric spaces. –  Hans Stricker Dec 20 '09 at 0:30
    
I don't know much about this stuff, but it looks like people who have commented on the original question suggest a wealth of resources for you to browse through... –  Aaron Mazel-Gee Dec 23 '09 at 1:47
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