We prove that UPC condition holds in o-minimal structures generated by some quasi-analytic classes of C ∞ functions. We also give a sufficient and necessary condition for a bounded set A ⊂ R 2 definable in some polynomially bounded o-minimal structure to be UPC.

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The aim of this paper is to prove the theorem on invariance of domain in an arbitrary o-minimal structure. We do not make use of the methods of algebraic topology and the proof is based merely on some basic facts about cells and cell decompositions.