Theorem: Chronology from a proper cone

For a proper cone $K\subset V$ of positive dimension, define

These relations have the stated causal-space properties, and

Every positive multiple of a nonzero boundary vector remains on the boundary.

Proof. Addition and pointedness give the partial order. The origin is not interior, since a neighborhood of zero would contain a nonzero vector and its negative.

If $a\in \mathrm{int}K$ and $b\in K$, choose $B(a,ϵ)\subset K$. Then $B(a+b,ϵ)\subset K+K\subset K$. Thus $\mathrm{int}K+K\subseteq \mathrm{int}K$, proving push-up. Closedness gives $K\setminus \mathrm{int}K=\partial K$. Positive dilation is a homeomorphism preserving both $K$ and its interior, hence its boundary.