The algebra of conditional sets and the concepts of conditional topology and compactness

摘要

The concepts of a conditional set, a conditional inclusion relation and aconditional Cartesian product are introduced. The resulting conditional settheory is sufficiently rich in order to construct a conditional topology, aconditional real and functional analysis indicating the possibility of amathematical discourse based on conditional sets. It is proved that theconditional power set is a complete Boolean algebra, and a conditional versionof the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, theBorel-Lebesgue theorem, the Hahn-Banach theorem, the Banach-Alaoglu theorem andthe Krein-\v{S}mulian theorem are shown.