Title
Stable topology and stable compactness
Abstract
In this paper we first develop the theory of stable sets with respect to a complete Boolean algebra by introducing some new set-theoretic tools. Then, we further propose the notions of a stable topological space and a stably compact space, and in particular focus on a systematic development of the theory of stably compact spaces. The resulting theory is very interesting: on the one hand, stable topological spaces are a class of topological spaces with a stable structure, their study requires an analysis of stratification structures coordinated with a partition of unity in a complete Boolean algebra; on the other hand, the theory of stable topological spaces and stably compact spaces also provides a considerable stable generalization of that of ordinary topological spaces and compact spaces, so that the theory of stably compact spaces can be used to deal with ubiquitous noncompactness problems. To show the power of stable compactness, we also demonstrate that most of compactness-based fundamental theorems established for locally convex spaces in functional analysis can be generalized to a wide class of (essentially non-locally convex) topological modules - random locally convex modules in random functional analysis in a stably compact manner, which considerably expands the applicable scope of the classical fundamental theorems. This is a recent work joint with Qiang Tu, Xiaohuan Mu, Goong Chen and Guang Yang.
