Connected components in d-minimal structures

Abstract

For a given d-minimal expansion $\mathfrak{R}$ of the ordered real field, we consider the expansion ${\mathfrak{R}}^{♮}$ of $\mathfrak{R}$ generated by the sets of the form ${\bigcup }_{S\in \mathcal{C}}S$, where $\mathcal{C}$ is a subfamily of the collection of connected components of an $\mathfrak{R}$-definable set. We prove that ${\mathfrak{R}}^{♮}$ is d-minimal. A similar assertion holds for almost o-minimal expansions of ordered groups.