Given a group (G, ·), G ⊆ M m , definable in a first order structure M = (M, . . .) equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V ⊆ G and define a new topology τ on G with which (G, ·) becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from M m . Likewise we topologize transitive group actions and fields definable in M. These results require a series of preparatory facts concerning dimension functions, some of which might be of independent interest.
IntroductionA. Pillay in [Pi2] adapts Hrushovski's unpublished proof of a special case of Weil's 'group chunk theorem' [W] to show that a group definable in an o-minimal structure can be definably equipped with a topology making it a topological group and a definable manifold. The topology in question on a large definable subset of the considered definable group coincides with the usual topology induced by ordering of the structure. It follows that a field definable in an o-minimal structure can be definably made into a topological field.As observed by A. Mosley in [Mo], Pillay's method works for groups and fields definable in sufficiently saturated first order topological structures (introduced in [Pi1]) on which certain very natural conditions have been imposed, like for example the exchange property of the algebraic closure. Among those topological structures we have several important classes of models satisfying various minimality conditions, namely: models of weakly o-minimal/C-minimal theories in which acl has the exchange property as well as P -minimal structures. Both [Pi2] and [Mo] extensively use the theory of generics. This paper contains an alternative approach towards the problem of topologization of definable algebraic objects. Instead of imposing the exchange property on acl, we consider topological structures equipped with so called dimension function.Such a context is related to that of [Mo] but does not coincide with it. For example, it is easy too see that if M is a weakly o-minimal structure and for any N M, acl has the exchange property, then the usual topological dimension (dim) is a dimension function in the sense of [vdD1]. On the other hand, if dim is a dimension function in a weakly o-minimal structure M, then acl has the exchange property in M (see [We06] for details).The paper is organized as follows. In §1 we fix some notation and discuss the basic notions that are used throughout the paper, like dimension functions and first order topological structures. In §2 we consider the issue of covering a group G definable in a first order structure equipped with a (weak) dimension function by a finite number of its translates. We give effective bounds on the number of translates of a large definable subset of G needed to cover G.Some of the preparatory results proved in §2 are used in §3- §5 to investigate groups, group actions and fields definable in first order topological structures equipped with a dimensi...