1. INTRODUCTION. In this paper we introduce the reader to two remarkable results in the theory of sets. Both are more than fifty years old, but neither one appears to be well known among nonspecialists. Each one states that a certain proposition implies the Axiom of Choice. First we describe the results, then review definitions, then, finally, present the proofs, most of which are straightforwa...

For a quantale V, the category V-Top of V-valued topological spaces may be introduced as a full subcategory of those V-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr’s characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for V completely distributive, V-topological ...

we introduce some new concepts of topological spaces which say α − separable topologicalspace and o-topological group, α − first axiom, α − second axiom, and we find some relations betweenthem with some applications in normed spaces.