Given a subset of X⊆Rn we can associate with every
point x∈Rn a vector space V of maximal dimension with the
property that for some ball centered at x, the subset X coincides inside
the ball with a union of lines parallel with V. A point is singular if V
has dimension 0. In an earlier paper we proved that a (R,+,<,Z)-definable relation X is actually definable in (R,+,<,1) if and only if the number of singular points is finite and every rational
section of X is (R,+,<,1)-definable, where a rational section is
a set obtained from X by fixing some component to a rational value. Here we
show that we can dispense with the hypothesis of X being (R,+,<,Z)-definable by assuming that the components of the singular points
are rational numbers. This provides a topological characterization of
first-order definability in the structure (R,+,<,1). It also
allows us to deliver a self-definable criterion (in Muchnik's terminology) of
(R,+,<,1)- and (R,+,<,Z)-definability for a
wide class of relations, which turns into an effective criterion provided that
the corresponding theory is decidable. In particular these results apply to the
class of k−recognizable relations on reals, and allow us to prove that it is
decidable whether a k−recognizable relation (of any arity) is
l−recognizable for every base l≥2.Comment: added sections 5 and 6, typos corrected. arXiv admin note: text
overlap with arXiv:2002.0428