According to Mack a space is countably paracompact if and only if its product with [0, 1] is δ-normal, i.e. any two disjoint closed sets, one of which is a regular Gδ-set, can be separated. In studying monotone versions of countable paracompactness, one is naturally led to consider various monotone versions of δ-normality. Such properties are the subject of this paper. We look at how these prop...

If X is a Tychonoff space, C(X) its ring of real-valued continuous functions. In this paper, we study non-essential ideals in C(X). Let a be a infinite cardinal, then X is called a-Kasch (resp. ā-Kasch) space if given any ideal (resp. z-ideal) I with gen (I) < a then I is a non-essential ideal. We show that X is an א0-Kasch space if and only if X is an almost P -space and X is an א1-Kasch space...

provided ƒ(2, x) is "sufficiently small." Our results, which are another contribution to this area, are motivated by two recent studies. Brauer and Wong [ l ] have obtained quite general results on the asymptotic relationships between the solutions of (1) and (2). We significantly weaken the hypotheses of one of their results; see Theorem 1 below. Toroshelidze [4] considered the problem of pert...

A Tychonoff space X is called RG if the embedding of C(X) → C(Xδ) is an epimorphism of rings. Compact RG spaces are known and easily described. We study the pseudocompact RG spaces. These must be scattered of finite Cantor Bendixon degree but need not be locally compact. However, under strong hypotheses, (countable compactness, or small cardinality) these spaces must, indeed, be compact. The ma...

If a locale is presented by a “flat site”, it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale (and also for its openness). Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies. A predicative proof of ...

O Lema Kuratowski-Zorn denominado frequentemente apenas por de Zorn é objeto controvérsias e debates desde o seu surgimento, apesar disto, sua relevância pode ser percebida pela grande quantidade aplicações consequências apresentadas sobre a forma enunciados equivalentes que são presentes na área exatas, também em matemáticas se baseiam teoria dos conjuntos como, axioma da Escolha, Teorema Boa ...

Abstract In the absence of axiom choice, set-theoretic status many natural statements about metrizable compact spaces is investigated. Some are provable in $$\mathbf {ZF}$$ ZF , some shown to be independent . For independence results, distinct models and permutation {ZFA}$$ ZFA with transfer ...

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...