Is there any research on the topic of sets of objects that don't allow for a definition of a distance between them? In other words, if we have some set of objects with some properties is there a way to prove that it is impossible to define any mathematical distance between them? [EDIT] a (trivial) distance of 1 can be always defined for distinct set elements so this part of the question doesn't have a substance.
Yet another way to ask this question is to say that there might be objects that we can not compare with each other in the way that is satisfying the norm equations (scalability, triangle, positive, zero). How can we prove that the set we have is such an "impossible" set? [EDIT] because a norm in this sense is defined on vector spaces, are there vector spaces that doesn't have any norm? Or is there also always some trivial norm possible? [EDIT2] indeed it is always possible to construct a norm as discussed in this question Is there such thing as an unnormed vector space?