Submission date: 11 February 2021

Abstract:

Given a subset of X ⊆ ℝ^{n} we can associate with every point x in ℝ^n 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 hyperplanes 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).

Mathematics Subject Classification: 03C64

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Full text arXiv 2102.06160: pdf, ps.