The Algebra of Conditional Sets and the Concepts of Conditional Topology and Compactness

Abstract

We introduce the notions of conditional set and conditional inclusion. It is shown that the resulting conditional power set is a complete Boolean algebra which allows to develop a set theory, topology and analysis on conditional sets. Among others a conditional version of the following theorems are proved: Ultrafilter lemma, Tychonoff, Heine-Borel, Hahn-Banach and Banach-Alaoglu.