Spaces determined by countably many locally compact subspaces

Preface: Volume 81

Preface: Volume 52, Issue 2

On the generalized spectra of topological algebras

The authors’ primary goal in this paper is to enhance the study of $T_0$ topological spaces by using the order of specialization of a $T_0$ -space to introduce the lower topology (with a subbasis of closed sets $\mathord{\uparrow } x$ ) and studying the interaction of the original topology and the lower topology. Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets $\mathord{\uparrow } F$ , $F$ finite, is contained in an open set $U$ , then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong $d$ -spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo $P$ with the Scott topology is a strong $d$ -space, then it is well-filtered, and if additionally the Scott topology of the product $P\times P$ is the product of the Scott topologies of the factors, then the Scott space of $P$ is sober. We also exhibit connections of this work with de Groot duality.

E. H. Spanier, Algebraic Topology, McGraw-Hill, New York (1966). S. B. Nadler, Jr., "Examples of fixed point free maps from cells onto larger cells and spheres," Rocky Mountain J. Math., ii, No. 2, 319-325 (1981). H. C. Miller, "On unicoherent continua," Trans. Am. Math. Soc., 69, No. i, 179-194 (1950). S. A. Bogatyi, "Periodic points of a mapping of a system of intervals of the line," V. Tiraspol Symposium on General Topology and Applications, 26-28, Kishinev: Shtiintsa (1985). A. N. Sharkovskii, "Coexistence of cycles of a continuous mapping of the line into itself," Ukr. Mat. Zh., 16, No. i, 61-71 (1964). B. Knaster, C. Kuratowski, and S. Mazurkiewicz, "Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe," Fund, Math., 14, 132-137 (1929).

Extreme amenability of abelian [formula omitted] groups

Introduction to topological groups

We study splittability over some classes of compact spaces which are useful in functional analysis and general topology. Among other things we show that a scattered pseudocompact space splittable over the class of Eberlein compact spaces is Eberlein compact. We also prove that a compact space splittable over the class of Eberlein compact spaces is hereditarily \(\sigma \)-metacompact, and that if X is a compact space splittable over the class of Corson compact spaces, then \(d(X)=w(X)\). We also obtain several results on Rosenthal and on descriptive compact spaces. For instance: (1) a compact space is Rosenthal if and only if it is splittable over the class of Rosenthal compacta, (2) \(\mathfrak {c}\)-metrizable countably compact spaces which split over the class of descriptive compacta are descriptive compact spaces, (3) if X is a compact space splittable over the class of descriptive compact spaces, then \(hd(X)=w(X)\), (4) scattered compact spaces splittable over the class of descriptive compact spaces are \(\sigma \)-discrete descriptive compacta, and (5) it is consistent with ZFC that a compact space which splits over the class of descriptive compacta is a descriptive compact space.

The main aim of the present paper is to define new soft separation axioms which lead us, first, to generalize existing comparable properties via general topology, second, to eliminate restrictions on the shape of soft open sets on soft regular spaces which given in [22], and third, to obtain a relationship between soft Hausdorff and new soft regular spaces similar to those exists via general topology. To this end, we define partial belong and total non belong relations, and investigate many properties related to these two relations. We then introduce new soft separation axioms, namely p-soft Ti-spaces (i = 0,1,2,3,4), depending on a total non belong relation, and study their features in detail. With the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft Ti-spaces are stronger than soft Ti-spaces, for i = 0,1,4. Also, we define a p-soft regular space, which is weaker than a soft regular space and verify that a p-soft regular condition is sufficient for the equivalent among p-soft Ti-spaces, for i = 0,1,2. Furthermore, we prove the equivalent among finite p-soft Ti-spaces, for i = 1,2,3 and derive that a finite product of p-soft Ti-spaces is p-soft Ti, for i = 0,1,2,3,4. In the last section, we show the relationships which associate some p-soft Ti-spaces with soft compactness, and in particular, we conclude under what conditions a soft subset of a p-soft T2-space is soft compact and prove that every soft compact p-soft T2-space is soft T3-space. Finally, we illuminate that some findings obtained in general topology are not true concerning soft topological spaces which among of them a finite soft topological space need not be soft compact.

On invariant tensors on the group ring of finite groups

5. Measure and Integration

We define the Riemann integral for bounded functions defined on a general topological measure space.When the space is a compact metric space the integral is equivalent to the R-integral defined by Edalat using domain theory. Introduction. Edalat[1] defined a Riemann type integral on a compact metric space, called the R-integral, using domain theory.The integral so defined has applications in various fields such as dynamic systems and chaos, and the work in [1] has also inspired other interesting research, see [2,3,5].The main properties of this new integral among others are: (1) If the space is [a, b], then this integral coincides with the ordinary Riemann integral;(2) a bounded function f is R-integrable if and only if it is continuous almost everywhere; (3) if f is R-integrable then it is also Lebesgue integrable and the value of the R-integral equals that of the Lebesgue integral of f .However, as the definition of the R-integral and most of the proofs in [1] rely heavily on very technical details of domain theory, this integral is hardly accessible to those who know little about domain theory.Furthermore, unlike the Riemann sum over a partition, the Riemann sum over a simple valuation, the key structure in defining the R-integral, lacks a clear geometric interpretation.In this paper we define a Riemann type integral with a domain-free approach.To make it easier to compare this integral with other known integrals we first introduce the more general M-integral for a given collection M of some measurable subsets satisfying certain conditions.The integral introduced here is defined for bounded real valued functions on an arbitrary topological measure space X which need not be a compact metric space as required in [1]; it is a generalization of the Riemann integral on intervals; a function f is integrable if and only if it is continuous almost everywhere when 2000

ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth's effectively given domains and Jung's strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott's domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.

The aim of this paper is to introduce the concept of soft compact spaces by using the notion of soft [Formula: see text]-open sets and soft [Formula: see text]-open covers, which we call soft [Formula: see text]-compact spaces. Many characterizations and properties of this space are obtained. Also, we discuss the concept of soft [Formula: see text]-compact subspaces and soft [Formula: see text]-compact subsets relative to the space.

15 - Introduction to topology