  We prove that if $X$ is a topological space that admits Debreu's classical
utility theorem (eg.\ $X$ is separable and connected, second countable, etc.),
then order relations on $X$ satisfying milder completeness conditions can be
continuously embedded in $\mathbb R^I$ for $I$ some index set. In the
particular case where $X$ is a compact metric space, this closes a conjecture
of Nishimura \& Ok (2015). We also show that when $\mathbb R^I$ is given a
non-standard partial order coinciding with Pareto improvement, the analogous
embedding theorem fails to hold in the continuous case.
