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arXiv · math/0307369

Topology of definable Hausdorff limits

Thierry Zell
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Abstract

Let $A\sub \R^{n+r}$ be a set definable in an o-minimal expansion $§$ of the real field, $A' \sub \R^r$ be its projection, and assume that the non-empty fibers $A_a \sub \R^n$ are compact for all $a \in A'$ and uniformly bounded, {\em i.e.} all fibers are contained in a ball of fixed radius $B(0,R).$ If $L$ is the Hausdorff limit of a sequence of fibers $A_{a_i},$ we give an upper-bound for the Betti numbers $b_k(L)$ in terms of definable sets explicitly constructed from a fiber $A_a.$ In particular, this allows to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative closure} to construct the o-minimal structure $§_\pfaff$ generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure $(X,Y)_0$ in the special case where $Y$ is empty.

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Thierry Zell. 2004-03-23. Topology of definable Hausdorff limits. https://doi.org/10.1007/s00454-004-1112-8

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