On the Non-sequential Nature of Domain Models of Real-number Computation

Escardó, Hofmann and Streicher showed that real-number computations in the interval-domain environment are inherently parallel, in the sense that they imply the presence of weak parallel-or. Part of the argument involves showing that the addition operation is not Vuillemin sequential. We generalize this to all continuous domain environments for the real line. The key property of the real line that leads to this phenomenon is its connectedness. We show that any continuous domain environment for any connected topological space exhibits a similar parallel effect.