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Hausdorff limits of one parameter families of definable sets in o-minimal structures

Kocel-Cynk B.

Czasopismo Techniczne. Nauki Podstawowe

2016

Y. 113, iss. 1-NP

73--80

We give an elementary proof of the following theorem on definability of Hausdorff limits of one parameter families of definable sets: let A [subset of] R×Rn be a bounded definable subset in o-minimal structure on (R,+,∙) such that for any y∈(0,c), c>0, the fibre Ay:={x∈Rn:(y,x)∈A} is a Lipschitz cell with constant L independent of y. Then the Hausdorff limit lim[y→0] Āy exists and is definable.

W prezentowanej pracy przedstawiamy elementarny dowód następującego twierdzenia o definiowalności granicy Hausdorffa jednoparametrowej rodziny zbiorów definiowalnych: niech A [pozdbiór] R×Rn będzie ograniczonym zbiorem definiowalnym w strukturze o-minimalnej typu (R,+,∙) takim, że dla dowolnego y∈(0,c), c>0, wókno Ay:={x∈Rn:(y,x)∈A} jest komórką Lipschitza ze staą L niezależną od y. Wtedy granica Hausdorffa lim[y→0] Āy istnieje i jest definiowalna.

The Whitney property of a fiber of a definable mapping

Kocel-Cynk B.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

141-147

We prove the following theorem Let S be a polynomially bounded o-minimal structure on (R,+,.) and let f : A --> [R^n] be a continuous, definable function on a compact definable set [A is subset of R^m]. Then there exist a positive real number [alpha belongs to R+] and a definable function C : f(A) --> R+ such that for any x [belongs to] f(A) and any two points p and q in the same connected component of [f^-1](x) there exists a piece-wise analytic curve gamma joinning p and q in [f^-1](x) with length [gamma is less than or equal to C(x)][...]. As a consequence we obtain the regular separation with parameter for definable sets.

Interior distance of two points in a fiber of a subanalytic map

Kocel-Cynk B.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

149-155

The aim of the paper is to prove the following theorem: Let N and M be two analytic manifolds and let K [is subset of] N be a compact subanalytic subset. For any continuous subanalytic map f : K --> M there exists a constant C > 0 such that for any y [belongs to] f(K) and any two points p, q in the same connected component of the fiber [f^-1](y) there exists a subanaiytic curve joining p and q in [f^-1](y) of length less than C.