  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 $(\mathbb
N,+,\cdot\,)$. We also show that any given expansion of $(\mathbb{R}, <,
+,\mathbb{N})$ by subsets of $\mathbb{N}^n$ ($n$ 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.
