We revisit monotonically semi-neighborhood refining (MSNR) spaces which were introduced by Stares in 1996. MSNR spaces are shown to be lob-spaces with well-ordered (F). The relationships between MSNR spaces with other monotone covering properties are also explored. We show the existence of MSNR spaces that do not posses a monotone locally-finite refining operator and spaces with a monotone locally-finite refining operator that are not MSNR answering a question of Popvassilev and Porter. Compact MSNR spaces may not be metrizable in general, but compact MSNR LOTS are. GO-spaces whose underlying LOTS has a sigma-closeddiscrete dense subset are shown to have a monotone star-finite refining operator. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A space has σ-compact tightness if the closures of σ-compact subsets determine the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known connections established between π-weight and the density of all dense subsets, or more precisely, the cardinal invariant δ(X).

We continue Gartside, Moody, and Stares' study of versions of monotone paracompactness. We show that the class of spaces with a monotone open closure-preserving operator is strictly larger than those with a monotone open locally-finite operator. We prove that monotonically metacompact GO-spaces have a monotone open star-finite operator, and so do GO-spaces with a monotone (open or not) closure-preserving operator, whose underlying LOTS has a σ-closed-discrete dense subset. A GO-space with a σ-closed-discrete dense subset and a monotone closure-preserving operator is metrizable. A compact LOTS with a monotone open closure-preserving operator is metrizable.

A topological space X is said to be e-separable if X has a sigma-closed-discrete dense subset. Recently, G. Gruenhage and D. Lutzer showed that e-separable PIGO spaces are perfect and asked if e-separable monotonically normal spaces are perfect in general. The main purpose of this article is to provide examples of e-separable monotonically normal spaces which are not perfect. Extremely normal e-separable spaces are shown to be stratifiable.

Properties weaker than monotone countable metacompactness are studied in PIGO and compact spaces. GO-spaces with a σ -closed-discrete dense subset and the N Z ( ω ) -property are metrizable generalizing results of Bennett, Hart and Lutzer and Peng and Li on monotonically countably metacompact spaces. NSR pair-families are introduced, and pair-bases that are NSR pair-families or countable unions of such pair-families are studied. By modifying results of Chase and Gruenhage, we show if a space X with a σ -NSR pair-base is compact, then X is metrizable, generalizing recent results by Chase and Gruenhage and some older results of Gruenhage and Nyikos. We show PIGO spaces with a σ -closed-discrete dense subset and a σ -NSR pair-base are metrizable. Relationships between these properties and others in the literature are also established.

We study the relationship between the Collins–Roscoe structuring mechanism and D-spaces. We introduce the notion of κ-sheltering and σ-sheltering point networks. We show that σ-sheltering (F) spaces are D-spaces which improves the results of Gruenhage and Yuming, and also show that Σ⁎-products of σ-sheltering (F) spaces are also σ-sheltering spaces. In addition, we show that ω1-sheltering (F) spaces are Dσ-spaces and transitively D-spaces. As a corollary, monotonically normal ω1-sheltering (F) spaces are paracompact.

Gartside and Moody proved that a space is protometrizable if and only if it has a monotone star-refinement operator on open covers. They called this property monotone paracompactness but noted that it might be better termed monotone full-normality, and posed the problem of characterizing spaces with a monotone locally-finite operator on open covers. Stares studied related monotone properties but left the above problem open. We introduce Nötherianly locally-finite bases, show that protometrizable spaces have such bases, and that spaces with such bases have a monotone locally-finite operator and are monotonically normal. An example of a non-protometrizable LOTS due to Fuller is shown to have a Nötherianly locally-finite base. The product L(ω1)×(ω+1) though not hereditarily normal, has a monotone locally-finite operator, while M×(ω+1) (where M is the Michael line) does not.

We define some monotone properties using stars of coverings. This relates to work of J. van Mill, V. Tkachuk, R. Wilson, O. Alas, L. Junqueira, M. Matveev and others who generalized the D-space property of E. van Douwen and E. Michael. Given a property P, we call a topological space X monotonically star-P if one can assign to each open cover U a subspace s(U)⊆X with property P in such a way that st(s(U),U)=⋃{U∈U:U∩s(U)≠∅}=X and if V refines U then s(U)⊆s(V). We study monotonically star-P spaces for various compactness-like properties P.

This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.

In 1951, Dowker proved that a space X is countably paracompact and normal if and only if X x I is normal. A normal space X is called a Dowker space if X x I is not normal. The main thrust of this article is to extend this work with regards alpha-normality and beta-normality. Characterizations are given for when the product of a space X and do (omega + 1) is alpha-normal or beta-normal. A new definition, acountably paracompact, illustrates what can be said if the product of X with a compact metric space is beta-normal. Several examples demonstrate that the product of a Dowker space and a compact metric space may or may not be alpha-normal or beta-normal. A collectionwise Hausdorff Moore space constructed by M. Wage is shown to be alpha-normal but not beta-nornal.

Since the pioneering work of Zadeh, fuzzy set theory has been applied to a myriad of areas. Song and Chissom introduced the concept of fuzzy time series and applied some methods to the enrollments of the University of Alabama. In recent years, a number of techniques have been proposed for forecasting based on fuzzy set theory methods. These methods have either used enrollment numbers or differences of enrollments as the universe of discourse. We propose using the year to year percentage change as the universe of discourse. In this communication, the approach of Jilani, Burney, and Ardil is modified by using the year to year percentage change as the universe of discourse. We use enrollment figures for the University of Alabama to illustrate our proposed method. The proposed method results in better forecasting accuracy than existing models. Keywords—Fuzzy forecasting, fuzzy time series, fuzzified enrollments, time-invariant model

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ1), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD(ℵ1) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω1.

This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.

This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.

The classical Helly's selection principle states that a uniformly bounded sequence of functions with uniform bounded variation admits a subsequence which converges pointwise to a function of bounded variation. Helly's selection principle for metric space-valued functions of bounded p-variation is proven answering a question of Chistyakov and Galkin.

A topological space is said to be totally paracompact if every open base of it has a locally finite subcover. It turns out this property is very restrictive. In fact, the irrationals are not totally paracompact. Here we give a generalization, base-paracompact, to the notion of totally paracompactness, and study which paracompact spaces satisfy this generalization.

THE outbreaks of dengue in the Caribbean area in 1963–64 and 1968–691 have served as reminders of the continuing presence of dengue in the Western Hemisphere, and the threat of recurrence of epidemic dengue in the southern United States. Since the first Caribbean pandemic in 1827, epidemics have been regularly reported in this region (Table 1). Their undiminished frequency, despite an increasing knowledge of causative viruses, disease vectors and means of prevention, must be taken as prima facie evidence of insufficient knowledge for effective long-term control or inadequate application of existing knowledge (or both). This paper will review the history . . .