[Maybe this is asking to be closed; but I thought I'd risk it.]
A metric satisfies the axioms:
- $d(x,y)=0$ if and only if $x=y$.
- $d(x,y) = d(y,x)$.
- $d(x,y) \leq d(x,z) + d(z,y)$.
Similarly (and motivationally) a uniformity on $X$ on a filter $F$ on $X\times X$ with:
- $\Delta = \{ (x,x) : x\in X \} \subseteq D$ for all $D\in F$.
- $D=\{ (y,x) : (x,y)\in D\}$ for all $D\in F$.
- for all $D\in F$ there is $E\in F$ with $E^2\subseteq F$.
It seems to me that if you drop the triangle-inequality (or the third axiom for a uniformity) then you can still do most basic topology, can still show, for example, that the space of bounded (uniformly) continuous functions $X\rightarrow\mathbb R$ or $\mathbb C$ is a Banach space (of interest to an Analyst like me) and so forth. My question is, why don't we study metrics/uniformities which doesn't satisfy the triangle inequality? (I see from Spaces with a quasi triangle inequality that such a thing is a semi-metric space).
Some reasons I thought of:
i) "Most" metrics arise from other structures– e.g. norms on a vector space– and the triangle-inequality comes from an axiom which seems more natural (e.g. the triangle-inequality for a norm really seems important– it gives some coupling between the additive structure of the vector space and the distance).
ii) If your space is quite nice (e.g. compact) in the topology induced by your semi-metric, then there is an actual metric giving the topology. So why not just assume you have a metric? This is especially true for uniform spaces, as you only need to be completely regular to be uniformisable.
iii) It seems to me that you really do need the triangle-inequality to talk about Cauchy-sequences in a sensible way, and so to talk about completeness.
I suppose I'm slightly motivated by the recent(ish) question on whether definitions in maths are "correct". What makes the triangle-inequality so useful that pretty much everyone assumes it, even though you can do a lot of point-set topology without it?
Best Answer
The triangle inequality is natural. In any setting where the metric is related to some kind of optimization problem, for example if $d(a, b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite abstractly, for example if $a$ and $b$ are states of some physical system and $d(a, b)$ describes the amount of energy that needs to be expended to get from $a$ to $b$), then $d(a, c) \le d(a, b) + d(b, c)$ because an optimal "path" from $a$ to $b$, together with an optimal "path" from $b$ to $c$, can be no better than the optimal "path" from $a$ to $c$.
It was Lawvere who first realized that the above sounds like composition of morphisms in a category, and this leads to Lawvere's definition of a metric space as a category enriched over the monoidal category $([0, \infty], +)$. From this perspective it's the other two axioms that aren't natural: the first axiom corresponds to ignoring isomorphisms, and the second axiom doesn't hold in some natural cases of the above argument (for example the "energy cost" metric won't always be symmetric). It's also not natural to require that metrics don't take the value $\infty$; this corresponds to no path existing between $a$ and $b$.
Perhaps the following example is useful. Given any graph $G$, there is a natural metric $d(a, b)$ given by the length of the shortest path from $a$ to $b$. If $G$ is a directed graph there is no reason to have $d(a, b) = d(b, a)$. On the other hand, given any graph $G$ we can construct the free category on its arrows.
(And if you're going to question why definitions in mathematics are correct, I find it curious that you'd question the triangle inequality before the definition of a topology.)