An achronal set that is not acausal: two null-related points

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain) for the general absolutely continuous causal-curve conclusions used here.

Given: The flat model $(A,\eta ,N)$ of The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets, written $A={\mathbb{R}}^4$, with its global inertial chart $x=(t,z)$, and the two events $p:=x^{-1}(0,0,0,0)$, $r:=x^{-1}(1,1,0,0)$.

For all events $u,v$ in the flat model, $u\le v$ exactly when $v-u\in \overline{C^{+}}$ and $u\ll v$ exactly when $v-u\in C^{+}$, with $C^{+}=\{w:w^0>\mathrm{\mid }{w}_s{\mathrm{\mid }}_{\mathrm{e}}\}$ and $\overline{C^{+}}=\{w:w^0\ge \mathrm{\mid }{w}_s{\mathrm{\mid }}_{\mathrm{e}}\}$ in the global inertial chart. For $u\mathrm{\ne }v$ the straight path realizes the relation when the displayed condition holds; for $u=v$, $u\le u$ instead follows from the equality alternative, without a constant causal curve (The flat model is globally hyperbolic and its inertial level sets are Cauchy hypersurfaces clause (a), The causal cones of the flat model in inertial coordinates, and closure of the future cone under sums clauses (a), (b), Time orientation, future and past timelike/null/causal vectors, and the closed cones, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

For events $u,v$ of the model: $u\ll v$ when there is a future timelike curve from $u$ to $v$, and $u\le v$ when $u=v$ or there is a future causal curve from $u$ to $v$; a set is achronal when no two distinct of its points are chronologically related and acausal when no two distinct of its points are causally related (Causal curves, chronological and causal reachability, and inextendibility clause (c), Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (b)).