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Is there a specific standard name for a mathematical space "$(\mathcal S, f)$"
consisting of a set $\mathcal S$ and a function $f : \mathcal{S \times S} \rightarrow \mathbb R$;
perhaps together with the ("obvious") additional property that $\forall A \in \mathcal S: f[~A, A~] = 0$
?

Some remarks:

The space "$(\mathcal S, f)$" I'm asking about presents a further generalization of the notion of a metric space "$(\mathcal M, d)$", including its various apparently more well-known generalizations since for all of them the function $d : \mathcal{M \times M} \rightarrow \mathbb R_{(\ge~0)}$, so they all involve the property $\forall A, B \in \mathcal M: d[~A, B~] \ge 0$.

In physics there's a well-known example of the kind of mathematical space I'm asking about, namely the space "$(\mathcal E, s^2)$", consisting of a (any suitable) set $\mathcal E$ of spacetime events and the function $s^2 : \mathcal{E \times E} \rightarrow \mathbb R$ expressing suitable spacetime intervals. This is of course known as (or at least closely related to) Minkowski space.

However, by definition of spacetime interval values $s^2$ they are applicable only to flat spacetime, i.e. with the additional property that for any 6 events $\in \mathcal E$ the corresponding Cayley-Menger determinant in terms of their 15 pairwise interval values vanishes. In other words, Minkowski space describes flat spacetime. But my question is concerned more generally with spaces which aren't necessarily flat.

user12262
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    Call such things however you want, picking a name which fits well with the context in which you want to use them. – Mariano Suárez-Álvarez Mar 12 '16 at 23:34
  • @Mariano Suárez-Alvarez: "Call such things however you want, picking a name which fits well with the context in which you want to use them." -- I like to know whether these "things" already have a specific standard name in the literature, in the hope that this enables me to find, learn from, and refer to the corresponding literature. Or are you suggesting that these "things", in all generality, have not been written about yet elsewhere? – user12262 Mar 12 '16 at 23:40

2 Answers2

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I don't think this type of space has its own field of study. Its too general. With just that condition on the function $f$ one could think of a thousand different structures without common features. Just choose any set $S$ define any function $\tilde{f}\colon S\times S\to\mathbb{R}$ and define $f\colon S\times S\to\mathbb{R}$ by $f(x,y)=\tilde{f}(x,y)$ if $x\neq y$ and $f(x,x)=0$.

How would you distinguish interesting behavour for such a space when so many ridiculous candidates could fit the bill?

Do you have any other conditions you can impose?

Aerinmund Fagelson
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  • Aerinmund Fagelson: "[...] too general. [...] one could think of a thousand different structures" -- Similar reservations might be held against the notion of a premetric, or correspondingly against a "premetric space". Nevertheless it has been considered and named, at least within the systematic consideration of generalizations of the notion "metric space". Then why not as well the further generalization I've been asking about. "Do you have any other conditions [to] impose?" -- At least not for the purpose of this question. – user12262 Mar 13 '16 at 00:19
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    @user12262 you are, of course, free to study whatever you like. If you accumulate enough results that merit the study worthy, great. Then it would be a good time to think of a suitable name for your structure. It is rather useless to spend some much time initially, before you have any justification for your study, in search of a name. Go ahead and investigate. Personally, I don't see the point in the study of a set together with a function as you describe. Perhaps you have some convincing reasons to explore these things. If so, what are they? Merely "others studied other things" is no reason. – Ittay Weiss Mar 13 '16 at 00:36
  • @Ittay Weiss: "rather useless to [...] initially seek a name" -- Well: I'd rather try to justify: "There's this already recognized/named mathematical structure of which I propose to study a special case", than: "I propose to study the following self-styled generalization of (or addition to) what's already recognized/named". At least, hereby, I've been exploring the former option. (Eventually I'd be looking to justify an idea of how sets of events which are described by pairwise Lorentzian d = 0 might be further differentiated.) – user12262 Mar 13 '16 at 02:18
  • Good luck!! If you can obtain interesting results about these 'spaces', then you can post some of your results and motivation, and seek ideas for a name. At this point, it's an unborn baby, one that was not even conceived yet. Don't get too attached to it. – Ittay Weiss Mar 13 '16 at 02:48
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In naming the described mathematical space "$(\mathcal S, f)$", along with property $\forall A \in \mathcal S: f[~A, A~] = 0$ it is useful and in accordance with existing terminology for generalized metric spaces to consider explicitly the property $\forall A, B \in \mathcal S : f[~A, B~] = f[~B, A~]$ as well.

The mathematical space "$(\mathcal S, f)$" consisting of a set $\mathcal S$ and a function $f : \mathcal{S \times S} \rightarrow \mathbb R$
(i.e. specificly without requiring non-negativity) but together with properties

(1): $\forall A \in \mathcal S: f[~A, A~] = 0$ (indiscernability of the identical), and

(2): $\forall A, B \in \mathcal S : f[~A, B~] = f[~B, A~]$ (symmetry)

can be called a "hypometric space".

Accordingly, the mathematical space as described in the question, with property (1) but without requiring property (2), would be referred to as a "quasihypometric space".

Requiring instead property (2), while dropping (1), would be called "metahypometric space"; etc.

user12262
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