Provided that the notion of "spacelike"-ness (of an interval) is symmetric: spacelike(x−y)⟺spacelike(y−x),
- of pairs containing the same element twice: IX:={x∈X:(xx)},
- of pairs of distinct elements whose interval is (called) "spacelike": SX:={x∈X&y∈X&x≠y&spacelike(x−y):(xy)}, and
- of all remaining pairs KX:=X2∖(IX∪SX).
Provided further that the notion of "lightlike"-ness (of an interval) is symmetric as well: lightlike(x−y)⟺lightlike(y−x),
- of pairs of distinct elements whose interval is (called) "lightlike": LX:={x∈X&y∈X&x≠y&lightlike(x−y):(xy)}, and
- of all remaining pairs TX:=KX∖LX ?
Edit
The wording of this question (apart from formatting issues) as it presently stands appears not adequate to the title. (Helpful responses to it have been received nevertheless, which I try to incorporate in going forward.)
In trying to improve the detailed wording, what would need to be considered first is (put roughly, for the time being, as I've come to consider it only recently):
(1) Whether and how the topological notion of "boundary" can be suitably generalized to the context of sets of pairs such as X2 and SX, and
(2) Whether, given a particular set X, the predicate "spacelike()" in the definition of set SX implies certain relations to a corresponding set KX which I did not state explicitly above; such as the absence of "impossible figures" wrt. membership of certain pairs in SX or KX.
I plan to defer editing (apart from possible formatting) until these preliminary questions have been expressed more adequatly elsewhere.
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