Cell Decomposition and Dimension Functions in First-order Topological Structures

The notion of a cell and that of a cell decomposition has been a central one in the study of certain first-order theories. A cell is a particular kind of definable set. The notion of a cell was first explicitly considered in [8], in the context of the theory of real closed fields. Collins defined a class of cells in this context, and showed that every definable subset of a real closed field is a finite disjoint union of cells. This property is often referred to as a cell decomposition for definable sets. The definable subsets of a real closed field are precisely the semi-algebraic sets (see [6]), and therefore this notion of a cell is of considerable importance in real algebraic geometry. Real closed fields are the principal examples of o-minimal structures. These have been extensively studied by model theorists during the last decade, and in this connection it was readily noticed that the notion of a cell was appropriate to the study of definable sets in this more general context. It was shown by van den Dries in [14] that o-minimal expansions of the real ordered field admit a cell decomposition for definable sets. However, Knight, Pillay and Steinhorn proved more generally in [19] that any o-minimal structure whatever admits a cell decomposition for definable sets. The notion of cell for o-minimal structures in [19] makes sense for linearly ordered structures, but a related notion was subsequently considered in a different context, that of the first-order theory of the p-adic numbers. Denef introduced a notion of cell appropriate to Qp in [10], and showed that any definable subset of Qp in the language of rings is a finite disjoint union of cells. By analogy with the real case, definable subsets of p-adically closed fields (structures elementarily equivalent to Qp) are often referred to as p-adic semi-algebraic sets. By contrast, however, a satisfactory analogue of an o-minimal structure in the p-adic setting has yet to be developed, much less an analogue of the cell decomposition for o-minimal structures in [19]. The structure of p-adic semi-algebraic sets was studied extensively by van den Dries and Scowcroft in [30]. They proved a result concerning the structure of definable subsets of Qp which, though weaker than the cell decomposition result in [10], is the basis for the work in this paper. We give a definition of cell (Definition 6.2) which makes sense in the context of any first-order structure in which there exists a definable topology, a first-order topological structure in the terminology of [24]. We also define in this context the Cell Decomposition