拓扑空间与度量结构的扰动
Topometric spaces and perturbations of metric structures
摘要：
我们发展了\emph{topometric spaces}的一般理论，即，拓扑空间配备了良好的下半连续度量函数。连续逻辑中的全局和局部类型的空间是研究这种空间的激励性例子。特别是，我们开发了一个理论的Cantor-Bendixson分析拓扑空间，它可以作为基础的研究的局部稳定性（扩展\textit{ad hoc}发展从\cite{BenYaacov-Usvyatsov:CFO}），以及全球$\aleph_0$稳定性。最后，我们研究扰动系统（见\cite{BenYaacov:Perturbations}）的拓扑空间的形式主义。特别是，我们展示了如何抽象的发展适用于$\aleph_0$稳定扰动。
英文摘要：
We develop the general theory of \emph{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 \textit{ad hoc} development from \cite{BenYaacov-Usvyatsov:CFO}), as well as of global $\aleph_0$-stability. We conclude with a study of perturbation systems (see \cite{BenYaacov:Perturbations}) in the formalism of topometric spaces. In particular, we show how the abstract development applies to $\aleph_0$-stability up to perturbation.
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