Connectedness in structures on the real numbers: o-minimality and undecidability

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o\nobreakdash-\hspace{0pt}minimal structures on $(\mathbb{R},<)$ have the property, as do all expansions of $(\mathbb{R},+,\cdot ,\mathbb{N})$. Our main analytic-geometric result is that any such expansion of $(\mathbb{R},<,+)$ by boolean combinations of open sets (of any arities) either is o\nobreakdash-\hspace{0pt}minimal or defines an isomorph of $(\mathbb N,+,\cdot,)$\mathrm{.}Wealsoshowthatanygivenexpansionof$($\mathbb{R}$, <, +,$\mathbb{N}$)$bysubsetsof$$\mathbb{N}$^n$($n$ allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.