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Expansions of o-minimal structures by sparse sets

100%

Friedman H., Miller C.

Fundamenta Mathematicae

2001

tom 167

nr 1

55-64

Given an o-minimal expansion ℜ of the ordered additive group of real numbers and E ⊆ ℝ, we consider the extent to which basic metric and topological properties of subsets of ℝ definable in the expansion (ℜ,E) are inherited by the subsets of ℝ definable in certain expansions of (ℜ,E). In particular, suppose that $f(E^{m})$ has no interior for each m ∈ ℕ and $f: ℝ^{m} → ℝ$ definable in ℜ, and that every subset of ℝ definable in (ℜ,E) has interior or is nowhere dense. Then every subset of ℝ definable in (ℜ,(S)) has interior or is nowhere dense, where S ranges over all nonempty subsets of all cartesian products $E^{k}$ (k ≥ 1). The same holds true with "nowhere dense" replaced by any of "null" (in the sense of Lebesgue), "countable", "a finite union of discrete sets", or "discrete". We use this (together with a result of L. van den Dries) to give an example of an expansion of the real field that defines an isomorphic copy of the ordered ring of integers, yet does not define ℤ.

Expansions of the real line by open sets: o-minimality and open cores

100%

Miller C., Speissegger P.

Fundamenta Mathematicae

1999

tom 162

nr 3

193-208

The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core of ℜ is o-minimal.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Miller C.

1 Friedman H.

1 Speissegger P.

1 2001

1 1999

Wyniki wyszukiwania

Expansions of o-minimal structures by sparse sets

Expansions of the real line by open sets: o-minimality and open cores