  In the context of combinatorial reciprocity, it is a natural question to ask
what "$-Q$" is for a poset $Q$. In a previous work, the definition
"$-Q:=Q\times\mathbb{R}$ with lexicographic order" was proposed based on the
notion of Euler characteristic of semialgebraic sets. In fact, by using this
definition, Stanley's reciprocity for order polynomials was generalized to an
equality for the Euler characteristics of certain spaces of increasing maps
between posets. The purpose of this paper is to refine this result, that is, to
show that these spaces are homeomorphic if the topology of $Q$ is metrizable.
