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

Abstract

The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel-Lebesgue theorem, the Hahn-Banach theorem, the Banach-Alaoglu theorem and the Krein-Šmulian theorem are shown.

29 pages

References in corpus (4)

• Recent progress in random metric theory and its applications to conditional risk measures
• A random version of Mazur's lemma
• Sheaves and conditional sets
• Conditional Analysis and a Principal-Agent problem

Cited by in corpus (13)

• An uncountable Furstenberg--Zimmer structure theory
• Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration
• Versions of Eberlein-Šmulian and Amir-Lindenstrauss theorems in the framework of conditional sets
• On compactness in $L^0$-modules
• Sheaves and conditional sets
• Measures and integrals in conditional set theory
• Vector duality via conditional extension of dual pairs
• A perturbational duality approach in vector optimization
• The noncompact Schauder fixed point theorem in random normed modules and its applications
• A Fenchel-Moreau theorem for ${\bar{L}}^0$-valued functions
• Brouwer Fixed Point Theorem in (L^0)^d
• A Boolean valued analysis approach to conditional risk
• A study of stability in locally $L^0$-convex modules and a conditional version of James' compactness theorem