[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?

aren'tmade CW. – Todd Trimble♦ May 20 '11 at 14:30