25,557 research outputs found
On chains in -closed topological pospaces
We study chains in an -closed topological partially ordered space. We give
sufficient conditions for a maximal chain in an -closed topological
partially ordered space such that contains a maximal (minimal) element.
Also we give sufficient conditions for a linearly ordered topological partially
ordered space to be -closed. We prove that any -closed topological
semilattice contains a zero. We show that a linearly ordered -closed
topological semilattice is an -closed topological pospace and show that in
the general case this is not true. We construct an example an -closed
topological pospace with a non--closed maximal chain and give sufficient
conditions that a maximal chain of an -closed topological pospace is an
-closed topological pospace.Comment: We have rewritten and substantially expanded the manuscrip
Topological set theories and hyperuniverses
We give a new set theoretic system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK and the theory of hyperuniverses. On the other hand, it retains most of the expressiveness of these theories and has the same consistency strength as ZF. We single out the additional axiom of the universal set as the one that increases the consistency strength to that of GPK and explore several other axioms and interrelations between those theories.
Hyperuniverses are a natural class of models for theories with a universal set. The Aleph_0- and Aleph_1-dimensional Cantor cubes are examples of hyperuniverses with additivity Aleph_0, because they are homeomorphic to their hyperspace. We prove that in the realm of spaces with uncountable additivity, none of the generalized Cantor cubes has that property.
Finally, we give two complementary constructions of hyperuniverses which generalize many of the constructions found in the literature and produce initial and terminal hyperuniverses
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