Monoton hücresel ayrışma

This thesis focuses on some results regarding the axioms of o-minimality which give rise to a restriction on definable sets ensuring the tameness of the topology, and also two regularity notions defined by van den Dries and by Gabrielov that we will simply call vdD-regularity which imposes some monotonicity properties on functions involved in the definition of open cells and definable functions, and Gabrielov-regularity which provides strong topological properties on cells. In this work, we first give some nice features of definable maps such as the Monotonicity Theorem and the Cell Decomposition Theorem which follow from the finiteness properties of definable sets in o-minimal structures. Then we present vdD-regularity which combines these two results and yields to state a more powerful theorem, namely Regular Cell Decomposition Theorem. In our work, we prove this theorem which has no proof in the literature. Afterwards, we are interested in a new regularity notion which is defined by Gabrielov to obtain cells that are topologically regular, whereas vdD-regularity does not necessarily imply this property. Indeed, we will examine an example in detail which is given by Gabrielov of a cell which is regular in the sense of van den Dries but not in the sense of Gabrielov.