UPC condition in polynomially bounded o-minimal structures
One of the purposes of this paper is to generalize the main result of \textit{W. Pawłucki} and \textit{W. Plésniak} in the paper [Math. Ann. 275, 467--480 (1986; Zbl 0579.32020)]. That is, the author proves that UPC condition holds in o-minimal structures generated by some quasi-analytic classes of \(C^\infty\) functions. We recall that UPC is the abbreviation for the terms `uniformly polynomially cuspidal' (the notion of UPC set was introduced by Pawłucki and Plésniak in the reference mentioned before). Moreover, there is given a characterization of bounded and plane UPC sets definable in polynomially bounded o-minimal structures. That is, if \(A\subseteq\mathbb R^2\) is bounded and definable in polynomially bounded o-minimal structures, then \(A\) is UPC if and only if \(A\) is fat (that is \(\overline A=\overline {\text{Int}(A)}\)) and the following condition is satisfied: for each \(a\in\overline A\), \(r>0\) and any connected component \(S\) of the set \(\text{Int}(A)\cap B(a,r)\) such that \(a\in\overline S\) there is a polynomial arc \(\gamma:(0,1)\to S\) such that \(\lim_{t\to 0}\gamma(t)=a\), where \(B(a,r)=\{x\in\mathbb R^2: \| x-a\|<r\}\).
scientific article; zbMATH DE number 2136631
scientific article