In this paper we will prove the Urysohn Meterization Theorem, which gives sufficient conditions for a topological space to be metrizable. In the process of proving this theorem, we will the discuss separation and countability axiom of topological spaces and prove the Urysohn Lemma.

This article continues the study of computable elementary topology started in [7]. We introduce a number of computable versions of the topological T0 to T3 separation axioms and solve their logical relation completely. In particular, it turns out that computable T1 is equivalent to computable T2. The strongest axiom SCT3 is used in [2] to construct a computable metric.

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.

Several axioms concerning probabilistic beliefs are examined here, and the relations between them are established, using belief spaces that generalize Harsanyi type spaces. Two axioms concerning highorder probabilistic beliefs are investigated in particular. The first is the triviality axiom, which says that one is certain (that is, has a belief of degree one) of one’s own beliefs. The second i...

Zermelo in his remarkable 1930a offered his final axiomatization of set theory as well as a striking, synthetic view of a procession of natural models that would have a modern resonance. Appearing only six articles after Skolem 1930 in Fundamenta mathematicae, Zermelo’s article seemed strategically placed as a response, an aspect that we will discuss below, but its dramatically new picture of s...

If the power set operation is considered as a definite operation, but the universe of all sets is regarded as an indefinite totality, we are led to systems of set theory having Power Set as an axiom but only Bounded Separation axioms and intuitionistic logic for reasoning about the universe at large. The study of subsystems of ZF formulated in intuitionistic logic with Bounded Separation but co...

we study different completeness definitions for two categories of lattice-valued cauchy spaces and the relations between these definitions. we also show the equivalence of a so-called completion axiom and the existence of a completion.