摘要:

ABSTRACT:IfXis an infinite set, we may topologize the power set ofXnaturally so that systems of subsets ofXcan be described in the language of Baire category. Systems we consider include: subalgebra lattices of algebraic structures; topologies; filters; and families of almost disjoint sets.

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44128.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:In this paper we study those uniform linearly ordered spaces that are determined by Pervin's quasi‐uniformity. In particular, we show that the space of all countable ordinals is determined by a unique quasi‐uniform

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44129.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:In this paper, we introduce the concepts ofp‐boundedness forpɛω*, (α,M)‐pseudocompactness and (α,M)‐compactness, for a cardinal number α and Ø≠M⊆β(ω)\ω. We prove thatXαis pseudocompact (respectively, countably compact) iffXis (α,M)‐pseudocompact (respectively, (α,M)‐compact), for some Ø≠M⊆β(ω)\ω; the Rudin‐Keisler order on β(ω)\ω can be defined in terms ofp‐boundedness andp‐pseudocompactness; and ifpɛβ(ω)\ω thenpisRK‐minimal (selective) iff the space ω∪T(p)i

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44130.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:It is known that a partially ordered set is uniquely representable, that is, it is order isomorphic to the order component of a Priestley space of a unique bounded distributive lattice whenever this lattice is generated by its doubly irreducible elements. In this case the interval topology on the poset is Hausdorff. Priestley has conjectured that the interval topology is always Hausdorff when the poset is uniquely representable. We consider this problem in topological terms. This allows us to see how the conjecture is a natural one and to broaden the class of lattices that are known to have uniquely representable spectral sets. Furthermore, we see that the theory of generalized continuous lattices ends up playing a crucial role, and that the conjecture perhaps should be strengthened to assert that for uniquely representable posets, the entire interval of admissible topologies collapses.

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44131.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:An ideal on a setXis a nonempty collection of subsets closed under the operations of finite union and subset. The concepts of parabounded and countably bounded subsets are defined as well as regularity with respect to an idealI(i.e.,I‐regular). The bounded subsets of a topological space are characterized as the subsets which are both parabounded and countably bounded. It is shown that if (X, τ) is a Hausdorff space andIis an ideal onXsuch thatI∩τ={Ø}, then (X, τ) is compact moduloIiff (X, τ) isH‐closed andI‐regular. Examples are given which placee‐Lindelöfness in the hierarchy of known properties and it is shown that para‐Lindelöfe‐Lindelöf spaces are Lindelöf. It is shown that in the category of Hausdorff spaces, perfect images ofe‐paracompact spaces aree‐paracompact. Posed as an open question is the following: What can be said about the closed continuous imag

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44132.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:K‐structures allow one to give purely set‐theoretic proofs of a number of important inequalities in cardinal functions. We explore the precise relationship between K‐structures, closure spaces in the sense of Čech, spaces with the weak topology in the sense of Arhangelskiǐ, and topological spaces. We also introduce the notion of a basic proximity space and establish the connection between K‐structures and basic proximit

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44133.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:We study the class of topologies which are induced by weighted quasi‐metrics (equivalently, partial metrics). Partial metrics were introduced by S. Matthews in his study of topological models appropriate for the denotational semantics of programming languages.It follows from our results that eachT0‐space with a s̀‐disjoint base admits a weightable quasi‐metric and that each weightable quasi‐metric space is quasi‐developable. Those partially ordered sets whose Alexandrov topology admits a weightable quasi‐metric are characterized. We also show that the Pixley‐Roy space over the reals does not admit a weighta

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44134.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:Given a poset (X; ≤), any spectral topology with specialisation order ≤ lies between the upper topology and the Scott topology. Conditions are given under which these extreme topologies are indeed spectral. The results contribute to the understanding ofspeectral sets, that is, posets which arise as the prime ideals of commutative rings with 1, ordered by inclusion, or equivalently as the prime ideals of distributive lattices with 0 and 1, ordered by inclus

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44135.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:The questions which motivated the development of neighborhood (nbhd) lattices as a generalization of topological (top) spaces are discussed. Nbhd systems, which are shown to be appropriate for characterizing continuity, are defined on ^‐semilattices, and are used to define open elements. The duals of nbhd systems are used to define closed elements in a lattice, independently of closure operators or complementation. In addition, the top continuity of a functionf:X→Yis characterized in terms of the nbhd continuity of the direct image function mappingP(X), the power set ofX, intoP(Y). T1‐nbhd lattices are defined, independently of points. Finally, the relationship between continuity and convergence is established by proving that a residuated function between conditionally completeT1‐nbhd lattices is continuous iff it preserves the limit of converge

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44136.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY

摘要:

ABSTRACT:We show that every regular quasi‐developable complete Aronszajn space is a subcompact space. From this result we deduce, among other things, that every regular quasidevelopable leftK‐sequentially complete quasi‐metric space is also subcompact. Finally, we prove that every Tychonoff Čech complete quasi‐metrizable space is a complete Aronsza

ISSN:0077-8923    

DOI:10.1111/j.1749-6632.1994.tb44137.x

出版商:Blackwell Publishing Ltd    

年代:1994

数据来源: WILEY