Author:
DOLICH ALFRED,MILLER CHRIS,SAVATOVSKY ALEX,THAMRONGTHANYALAK ATHIPAT
Abstract
AbstractWe initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o-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-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.
Publisher
Cambridge University Press (CUP)