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Is there a theory of metric spaces in which the distance between a given pair of points need not be defined?

I'm aware that there is a theory of partial metric spaces, but these deal with a different generalization of metric spaces, viz. ones in which $d(x,x)=0$ does not necessarily hold.

What I'm interested in here is the case where the metric axioms hold as far as all distances appearing are defined (I assume that $d(x,x)=0$ is always defined, and $d(y,x)=d(x,y)$ is defined if $d(x,y)$ is defined), but there may be pairs of points for which the distance is not defined. (A trivial example would be a disjoint union of two metric spaces.) Is there a theory of such structures?

In particular I'm interested in conditions under which the partially-defined metric can be consistently extended into a metric.

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    $\begingroup$ I don't remember the name, but there's an obvious notion where $\infty$ distance is allowed (with the obvious axiom: triangle inequality holding for all triples). Examples are given by the geodesic distance between vertices of a possibly non-connected graph. $\endgroup$
    – YCor
    Commented Mar 9, 2021 at 10:10
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    $\begingroup$ Of course with no further axiom, there are very different examples: for instance the plane, where the Euclidean distance is defined whenever it lies in some give subset of the reals. $\endgroup$
    – YCor
    Commented Mar 9, 2021 at 10:13
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    $\begingroup$ (The notion you refer of "partial metric space", to mean self-distances can be positive, has a very awkward choice of terminology, used by very few apparently and hence which could be safely discarded.) $\endgroup$
    – YCor
    Commented Mar 9, 2021 at 10:15
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    $\begingroup$ I think you are looking for so called "extended metric spaces". They are indeed metrics where infinite distances are allowed as YCor already mentioned. They are not much different from ordinary metric spaces as discussed here: math.stackexchange.com/questions/1964378/… $\endgroup$
    – Bert Lindenhovius
    Commented Mar 9, 2021 at 10:36
  • $\begingroup$ I think the concept I'm looking for is more general than extended metric spaces, since the case of undefined distance need not comply with a triangle inequality. I'm thinking of something like a cycle (a,b,c,d) with d(a,b)=d(b,c)=d(c,d)=1, d(a,d)=9, and d(a,c), d(b,d) undefined. I don't think these latter two can be infinite and/or obey the triangle inequality. $\endgroup$
    – gmvh
    Commented Mar 9, 2021 at 12:42

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It seems that one can in many cases construct an extended metric space from what I'll call a merometric space $X$ (i.e. the kind of partially defined metric space considered in the question) by proceeding as follows:

  • Call a finite sequence $\gamma$ of points $\gamma_i\in X$ a polygonal path if the distances $d(\gamma_i,\gamma_{i+1})$ are all defined.
  • Assign to every polygonal path a length $\ell(\gamma)=\sum_i d(\gamma_i,\gamma_{i+1})$.
  • Set the distance $D(x,y)=\inf_{\gamma,\gamma_0=x,\gamma_N=y} \ell(\gamma)$.

If there are no polygonal paths from $x$ to $y$, $D(x,y)=\infty$, so this is an extended metric. If $d(x,y)$ is defined, $D(x,y)\le d(x,y)$, so there is a reasonable sense in which $D$ completes $d$. The triangle inequality and symmetry are trivially fulfilled. The only obstruction that can arise is that in the presence of an infinite number of polygonal paths this construction can yield $D(x,y)=0$ for $x\not=y$, in which case $D$ isn't a metric. I haven't been able to figure out a necessary and sufficient condition that prevents this.

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