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Definable continuous selections of set-valued maps in o-minimal expansions of the real field

Sokantika S., Thamrongthanyalak A.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

Vol. 65, no. 2

97--105

Let T be a set-valued map from a subset of Rn to Rm. Suppose (R;+,⋅,T) is o-minimal. We prove that (1) if for every x ∈ Rn, each connected component of T(x) is convex, then T has a continuous selection if and only if T has a continuous selection definable in (R;+,⋅,T); (2) if n = 1 or m = 1, then T has a continuous selection if and only if T has a continuous selection definable in (R;+,⋅,T).

On Intersections of Generic Perturbations of Definable Sets

Mycielski J., Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 2/3

95--103

Consider an o-minimal expansion R of a real closed field R and two definable sets E and M. We introduce concepts of a locally transitive (abbreviated to l.t.) and a strongly locally transitive (abbreviated to s.l.t.) action of E on M. In the former case, M is supposed to be of pure dimension m; in the latter, both M and E are supposed to be of pure dimension. We treat the elements of E as perturbations of the set M. We prove that if E acts l.t. on M, and A and B are two non-empty definable subsets of M of dimension dim A≤ dim B < dim M, then dim(σ(A) ∩ B) < dim A for a generic σ in E; here dim ∅ = −1. And if E acts s.l.t. on M and A and B are two definable subsets of M, then dim(σ(A) ∩ B) ≤ max{dim A + dim B − m, −1} for a generic σ in E. We give an example of a l.t. action E on M for which the latter conclusion of the intersection theorem fails. We also prove a theorem on the intersections of generic perturbations in terms of the exceptional set T ⊂ M of points at which E is not l.t. Finally, we provide some natural conditions which imply that T is a nowhere dense subset of M.