Groups, group actions and fields definable in first-order topological structures

Given a group (G, ·), G ⊆M, definable in a first order structureM = (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. 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.