11 research outputs found
G-Compactness and Groups
Lascar described E_KP as a composition of E_L and the topological closure of
EL. We generalize this result to some other pairs of equivalence relations.
Motivated by an attempt to construct a new example of a non-G-compact theory,
we consider the following example. Assume G is a group definable in a structure
M. We define a structure M_0 consisting of M and X as two sorts, where X is an
affine copy of G and in M_0 we have the structure of M and the action of G on
X. We prove that the Lascar group of M_0 is a semi-direct product of the Lascar
group of M and G/G_L. We discuss the relationship between G-compactness of M
and M_0. This example may yield new examples of non-G-compact theories.Comment: 18 page
TOPOMETRIC SPACES AND PERTURBATIONS OF METRIC STRUCTURES
We develop the general theory of topometric spaces, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric function. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop a theory of Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the ad hoc development from [BU]), as well as of global ℵ0-stability. We conclude with a study of perturbation systems (see [Benb]) in the formalism of topometric spaces. In particular, we show how the abstract development applies to ℵ0-stability up to perturbation