general relativity - If it is given which intervals are spacelike, can be determined which intervals are lightlike?

Provided that the notion of "$\mbox{spacelike}$"-ness (of an interval) is symmetric: $$\text{spacelike}( \, x - y \, ) \Longleftrightarrow \text{spacelike}( \, y - x \, ),$$ then for any set $X$ (of sufficiently many elements) the set $X^2$ (of all pairs, regardless of order, of not necessarily distinct elements of $X$) may be partitioned into three disjoint and generally non-empty subsets

• of pairs containing the same element twice: $I_X := \{ x \in X: (x x) \}$,


• of pairs of distinct elements whose interval is (called) "$\text{spacelike}$": $S_X := \{ x \in X \, \& \, y \in X \, \& \, x \ne y \, \& \, \text{spacelike}( x - y ): (x y) \}$, and


• of all remaining pairs $K_X := X^2 \backslash (I_X \cup S_X)$.

Provided further that the notion of "$\mbox{lightlike}$"-ness (of an interval) is symmetric as well: $$\text{lightlike}( \, x - y \, ) \Longleftrightarrow \text{lightlike}( \, y - x \, ),$$ and given a suitable set $X$ (of sufficiently many elements) and a suitable set $S_X$ satisfying $X^2 \cap S_X = S_X$, how would the corresponding set $K_X$ be partitioned further into two disjoint and generally non-empty subsets

• of pairs of distinct elements whose interval is (called) "$\text{lightlike}$": $L_X := \{ x \in X \, \& \, y \in X \, \& \, x \ne y \, \& \, \text{lightlike}( x - y ): (x y) \}$, and


• of all remaining pairs $T_X := K_X \backslash L_X$ ?

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 $X^2$ and $S_X$, and

(2) Whether, given a particular set $X$, the predicate "$spacelike()$" in the definition of set $S_X$ implies certain relations to a corresponding set $K_X$ which I did not state explicitly above; such as the absence of "impossible figures" wrt. membership of certain pairs in $S_X$ or $K_X$.

I plan to defer editing (apart from possible formatting) until these preliminary questions have been expressed more adequatly elsewhere.