Core Compact Spaces Research Articles

Maximal classes of spaces and domains determined by topologies on function spaces of domains

In this paper, inspired by the work of Lawson and Xu, we consider the question of what classes A of topological spaces should be paired with classes B of domains in order that the Isbell and Scott topologies on function spaces [A→B] are identical for A∈A and B∈B. It is shown that: (1) if A and B are restricted to be a class of core compact spaces and a class of bicomplete domains respectively, a class of core compact and compact spaces is paired with a class of conditionally bounded complete domains; (2) if A and B are restricted to be a class of core compact and locally connected spaces and a class of bicomplete domains respectively, a class of topological spaces called RW-spaces (with finitely many connected components), which was first introduced by Kou and Luo and further studied by Lawson and Xu later, is paired with a class of pointed L-domains (disjoint union of pointed L-domains). This also gives a solution to a problem posed by Kou and Luo.

Function spaces from coherent continuous domains to RB-domains

In this paper, continuing the work of the first and third authors, we study the function spaces from coherent continuous domains to RB-domains. Firstly, we prove that the function spaces from coherent core compact spaces with compact open sets as a basis to bifinite domains (algebraic RB-domains) are algebraic. As an application, we give an example which shows that a function space from a coherent quasi-algebraic domain to a finite domain might not be coherent. Finally, we show that the function space from a coherent continuous domain to an RB-domain is an RB-domain. Particularly, a function space from an FS-domain (introduced by Achim Jung) to an RB-domain is an RB-domain. This addresses an old open problem of whether FS-domains are RB-domains.

Coincidence of the Isbell and Scott topologies on domain function spaces

Let [X→L] be the set of all continuous mappings from a topological space X to a domain L with the pointwise order. Let Is[X→L], σ[X→L] be the Isbell and Scott topologies on [X→L] respectively. In this paper, the question of when the Isbell and Scott topologies coincide on [X→L] is considered. Main results are:(1)If L is a bicomplete domain, then (i) Is[X→L]=σ[X→L] for all core compact spaces X if and only if L is bounded complete; (ii) Is[X→L]=σ[X→L] for all core compact and compact spaces X if and only if L is conditionally bounded complete.(2)Let L be a dcpo consisting of a least element and a decreasing sequence with two lower bounds, then Is[X→L]=σ[X→L] for all core compact spaces X with a countable base.(3)Let X be a well rooted Lawson compact domain and L an FL-domain, then Is[X→L]=σ[X→L].

On the compact-open and admissible topologies

It is known (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101–119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105–145; D.N. Georgiou, S.D. Iliadis, F. Mynard, Function space topologies, in: Open Problems in Topology 2, Elsevier, 2007, pp. 15–23]) that the intersection of all admissible topologies on the set C ( Y , Z ) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology (which in general is not admissible). The following, interesting in our opinion, problem is arised: when a given splitting topology (for example, the compact-open topology, the Isbell topology, and the greatest splitting topology) is the intersection of k admissible topologies, where k is a finite number. Of course, in this case this splitting topology will be the greatest splitting. In the case, where a given splitting topology is admissible the above number k is equal to one. For example, if Y is a locally compact Hausdorff space, then k = 1 for the compact-open topology (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429–432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480–495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31]). Also, if Y is a corecompact space, then k = 1 for the Isbell topology (see [P. Lambrinos, B.K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, in: Continuous Lattices and Applications, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 191–211; F. Schwarz, S. Weck, Scott topology, Isbell topology, and continuous convergence, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 251–271]). In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31] a non-locally compact completely regular space Y is constructed such that the compact-open topology on C ( Y , S ) , where S is the Sierpinski space, coincides with the greatest splitting topology (which is not admissible). This fact is proved by the construction of two admissible topologies on C ( Y , S ) whose intersection is the compact-open topology, that is k = 2 . In the present paper improving the method of [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31] we construct some other non-locally compact spaces Y such that the compact-open topology on C ( Y , S ) is the intersection of two admissible topologies. Also, we give some concrete problems concerning the above arised general problem.

A generalization of core compact spaces

In this paper, for a fixed infinite cardinal ν, we give the notion of a ν-core compact space generalizing the notion of a core compact space. Also, on the family of all open subsets of a space Y (actually on a partially ordered set) we define a topology denoted by τ ν and calling ν-Scott topology. This topology defines on the set C ( Y , Z ) of all continuous functions of the space Y into a space Z a topology denoted by t ν . Some relations between ν-core compact spaces and topologies τ ν and t ν are given.

An Elementary Approach to Exponentiable Maps

Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.

Comparing Cartesian closed categories of (core) compactly generated spaces

It is well known that, although the category of topological spaces is not Cartesian closed, it possesses many Cartesian closed full subcategories, e.g.: (i) compactly generated Hausdorff spaces; (ii) quotients of locally compact Hausdorff spaces, which form a larger category; (iii) quotients of locally compact spaces without separation axiom, which form an even larger one; (iv) quotients of core compact spaces, which is at least as large as the previous; (v) sequential spaces, which are strictly included in (ii); and (vi) quotients of countably based spaces, which are strictly included in the category (v). We give a simple and uniform proof of Cartesian closedness for many categories of topological spaces, including (ii)–(v), and implicitly (i), and we also give a self-contained proof that (vi) is Cartesian closed. Our main aim, however, is to compare the categories (i)–(vi), and others like them. When restricted to Hausdorff spaces, (ii)–(iv) collapse to (i), and most non-Hausdorff spaces of interest, such as those which occur in domain theory, are already in (ii). Regarding the Cartesian closed structure, finite products coincide in (i)–(vi). Function spaces are characterized as coreflections of both the Isbell and natural topologies. In general, the function spaces differ between the categories, but those of (vi) coincide with those in any of the larger categories (ii)–(v). Finally, the topologies of the spaces in the categories (i)–(iv) are analyzed in terms of Lawson duality.

Posets having continuous intervals

In this paper we consider posets in which each order interval [ a, b] is a continuous poset or continuous domain. After developing some basic theory for such posets, we derive our major result: if X is a core compact space and L is a poset equipped with the Scott topology (assumed to satisfy a mild extra condition) for which each interval is a continuous sup-semilattice, then the function space of continuous locally bounded functions from X into L has intervals that are continuous sup-semilattices. This substantially generalizes known results for continuous domains.

Maximal Classes of Topological Spaces and Domains Determined by Function Spaces

In this paper the question of what classes A of T 0-spaces should be paired with classes ℬ of domains in order that all function spaces [A→B] for A∈A and B∈ℬ are λ-compact domains is considered. It is shown that core compact spaces are paired with bounded complete domains and a class of topological spaces called RW-spaces (with finitely many components) is paired with the class of λ-compact pointed L-domains (L-domains).

When does the class [formula omitted] consist of continuous domains?

Given classes of domains (or topological spaces) A and B , when are all function spaces [ A→ B] again continuous domains? The principle result of this paper is that for A either all compact and core compact spaces or only the single domain consisting of a decreasing sequence with two lower bounds, then the largest B consists of all continuous domains such that ↓ x is a sup-semilattice for each x. We also establish an analogue for L-domains.

RW-spaces and compactness of function spaces for L-domains

The purpose of this paper is to study the Lawson compactness of function spaces for L-domains. A basic notion of property RW for core compact spaces is introduced, which is proved to have a close relation to the Lawson compactness of function spaces for continuous L-domains as following: (1)Every Lawson compact continuous dcpo has property RW (via the Scott topology) and for each continuous L-domain, Lawson compactness is equivalent to property RW;(2)Let P be a continuous dcpo with a least element. Then [X→P] is compact continuous for every core compact space X with property RW iff P is compact continuous L-domain;(3)Let X be a core compact space. Then [X→P] is compact for every compact continuous L-domain P iff X has property RW.

Solutions to two problems of J.D. Lawson and M. Mislove

The problem of characterizing those L for which [ X → L] is a continuous DCPO for all core compact spaces is solved. Also, the problem of coincidence of the Isbell topology with the Scott topology on [ X → L] is solved.

Sobrification and bicompletion of totally bounded quasi-uniform spaces

AbstractWe observe that if is a compatible totally bounded quasi-uniformity on a T0-space (X,), then the bicompletion of (X, ) is a strongly sober, locally quasicompact space. It follows that the b-closure S of (X, ) in is homeomorphic to the sobrification of the space (X, ). We prove that S is equal to if and only if (X, ) is a core-compact space in which every ultrafilter has an irreducible convergence set and is the coarsest quasi-uniformity compatible with . If is the Pervin quasi-uniformity on X, then S is equal to if and only if X is hereditarily quasicompact, or equivalently, is the Pervin quasi-uniformity on .