Bispaces admitting only bicomplete or only totally bounded quasi-metrics

We characterize quasi-metrizable bispaces that admit only bicomplete quasimetrics by means of doubly primitive sequences, and deduce that if (X, S, T) is a quasi-metrizable bispace admitting only bicomplete quasi-metrics and either (X, S) or (X, T) is hereditarily Lindelöf, then (X, S ∨ T) is compact. We also give an example which shows that hereditary Lindelöfness cannot be omitted in the above result. Finally, we show that a quasi-pseudometrizable bispace (X, S, T) admits only totally bounded quasi-pseudometrics if and only if (X, S ∨ T) is compact, and deduce that a quasi-pseudometrizable topological space admits only totally bounded quasi-pseudometrics if and only if it is hereditarily compact and quasi-sober (equivalently, if and only if it admits a unique quasi-uniformity).     

Totally Bounded Quasi-Uniformities and Pseudocompactness of Bispaces

 We characterize pairwise Tychonoff bispaces that admit only totally bounded quasi-uniformities in terms of a suitable notion of bitopological pseudocompactness. We also show that a pairwise Tychonoff bispace has a unique (up to equivalence) bicompactification if and only if it admits a unique totally bounded quasi-unifomity. These results extend classical theorems of R. Doss for uniform spaces to the quasi-uniform (bitopological) setting, and are applied to the study of T ₀ topological spaces that admit a unique quasi-uniformity and a unique bicompactification, respectively. Finally, we discuss the problem of extending the classical Glicksberg theorem on product of pseudocompact spaces to bispaces and a partial solution is obtained. Supported by the Spanish Ministry of Science and Technology, grant BFM2000-1111. Supported by a grant from Generalitat Valenciana. Received November 7, 2001; in revised form August 14, 2002     

STRONG ZERO-DIMENSIONALITY OF BIFRAMES AND BISPACES

Abstract

A bispace is called strongly zero-dimensional if its bispace Stone—?ech compactification is zero—dimensional. To motivate the study of such bispaces we show that among those functorial quasi—uniformities which are admissible on all completely regular bispaces, some are and others are not transitive on the strongly zero-dimensional bispaces. This is in contrast with our result that every functorial admissible uniformity on the completely regular spaces is transitive precisely on the strongly zero-dimensional spaces.

We then extend the notion of strong zero-dimensionality to frames and biframes, and introduce a De Morgan property for biframes. The Stone—Cech compactification of a De Morgan biframe is again De Morgan. In consequence, the congruence biframe of any frame and the Skula biframe of any topological space are De Morgan and hence strongly zero-dimensional. Examples show that the latter two classes of biframes differ essentially.     

The bicompletion of the Hausdorff quasi-uniformity

We study conditions under which the Hausdorff quasi-uniformity UH of a quasi-uniform space (X,U) on the set P₀(X) of the nonempty subsets of X is bicomplete.Indeed we present an explicit method to construct the bicompletion of the T₀-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform T₀-spaces (X,U) for which the Hausdorff quasi-uniformity of their bicompletion on is bicomplete.     

Bitopologies on Totally Ordered Sets

We study the category of ray bispaces, that is, the category whose objects are totally ordered sets with two topologies, each having a subbase of rays and so that the resulting bitopological space is pairwise weakly symmetric, and whose morphisms are the pairwise continuous functions. In contrast with the purely topological results of [5], we show that, (1) such spaces are utterly normal and hence monotonically normal (in the sense of [6]), and (2) (Intermediate Value Theorem) the pairwise continuous image of a pairwise connected bitopological space in a selective ray bispace is an interval. We also obtain conditions for the equality of the de Groot dual (see [4]) and the ray dual (see [5]) of a ray topology and show that a selective ray topology is compact if and only if it is skew compact.     

PAIRWISE ALMOST COMPACT SPACES

ABSTRACT

The purpose of this paper is to investigate pairwise almost compact bitopological spaces. These spaces satisfy a bitopological compactness criterion which is strictly weaker than pairwise C-compactness and is independent of other well-known bitopological compactness notions. Pairwise continuous maps from such spaces to pairwise Hausdorff spaces are pairwise almost closed, the property is invariant under suitably continuous maps, is inherited by regularly closed subspaces and may be characterized in terms of certain covers as well as the adherent convergence of certain open filter bases. Some new natural bitopological separation axioms are introduced and in conjunction with pairwise almost compactness yield interesting results, including a sufficient condition for the bitopological complete separation of disjoint regularly closed sets by semi-continuous functions.     

The β-space property in monotonically normal spaces and GO-spaces

In this paper we examine the role of the β-space property (equivalently of the MCM-property) in generalized ordered (GO-)spaces and, more generally, in monotonically normal spaces. We show that a GO-space is metrizable iff it is a β-space with a Gδ-diagonal and iff it is a quasi-developable β-space. That last assertion is a corollary of a general theorem that any β-space with a σ-point-finite base must be developable. We use a theorem of Balogh and Rudin to show that any monotonically normal space that is hereditarily monotonically countably metacompact (equivalently, hereditarily a β-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a β-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos.     

More on continuously Urysohn spaces

We study the properties of weakly continuously Urysohn and continuously Urysohn spaces. We show that being a (weakly) continuously Urysohn space is not a multiplicative property, and that this property is not preserved under perfect maps. However, being a weakly continuously Urysohn space is preserved under perfect open maps. By using the scattering process, we show that the class of protometrizable spaces is also contained in the class of continuously Urysohn space. We also give a characterization of the continuously Urysohn property for well-ordered spaces, and prove that a paracompact locally continuously Urysohn ordered space is continuously Urysohn.     

Monotone versions of δ-normality

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 properties relate to each other and prove a number of results about them, in particular, we provide a factorization of monotone normality in terms of monotone δ-normality and a weak property that holds in monotonically normal spaces and in first countable Tychonoff spaces. We also discuss the productivity of these properties with a compact metrizable space.     

BIFRAMES AND BISPACES

Abstract

The concept of a biframe is introduced. Then the known dual adjunction between topological spaces and frames (i.e. local lattices) is extended to one between bispaces (i.e. bitopological spaces) and biframes. The largest duality contained in this dual adjunction defines the sober bispaces, which are also characterized in terms of the sober spaces. The topological and the frame-theoretic concepts of regularity, complete regularity and compactness are extended to bispaces and biframes respectively. For the bispaces these concepts are found to coincide with those introduced earlier by J.C. Kelly, E.P. Lane, S. Salbany and others. The Stone-?ech compactification (compact regular coreflection) of a biframe is constructed without the Axiom of Choice.     

More about complements of quasi-uniformities

We continue our investigations on the lattice (q(X),⊆) of quasi-uniformities on a set X. Improving on earlier results, we show that the Pervin quasi-uniformity (resp. the well-monotone quasi-uniformity) of an infinite topological T₁-space X does not have a complement in (q(X),⊆). We also establish that a hereditarily precompact quasi-uniformity inducing the discrete topology on an infinite set X does not have a complement in (q(X),⊆).     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

Infima and complements in the lattice of quasi-uniformities

We show that the infimum of any family of proximally symmetric quasi-uniformities is proximally symmetric, while the supremum of two proximally symmetric quasi-uniformities need not be proximally symmetric. On the other hand, the supremum of any family of transitive quasi-uniformities is transitive, while there are transitive quasi-uniformities whose infimum with their conjugate quasi-uniformity is not transitive. Moreover we present two examples that show that neither the supremum topology nor the infimum topology of two transitive topologies need be transitive. Finally, we prove that several operations one can perform on and between quasi-uniformities preserve the property of having a complement.     

Permutable pairs of quasi-uniformities

We continue investigating the lattice (q(X),⊆) of quasi-uniformities on a set X. In particular in this article we start investigating permutable pairs of quasi-uniformities. Among other things, we show that the Pervin quasi-uniformity of a topological space X permutes with its conjugate if and only if X is normal and extremally disconnected.     

Infima of quasi-uniform anti-atoms

Let (q(X),⊆) denote the lattice consisting of the set q(X) of all quasi-uniformities on a set X, ordered by set-theoretic inclusion ⊆. We observe that a quasi-uniformity on X is the supremum of atoms of (q(X),⊆) if and only if it is totally bounded and transitive. Each quasi-uniformity on X that is totally bounded or has a linearly ordered base is shown to be the infimum of anti-atoms of (q(X),⊆). Furthermore, each quasi-uniformity U on X such that the topology of the associated supremum uniformity Us is resolvable has the latter property.     

A note on monotonically metacompact spaces

We show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO) spaces. We show, for example, that a generalized ordered space with a σ-closed-discrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S.G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.     

A note on monotone covering properties

In this note, we show that a monotonically normal space that is monotonically countably metacompact (monotonically meta-Lindelöf) must be hereditarily paracompact. This answers a question of H.R. Bennett, K.P. Hart and D.J. Lutzer. We also show that any compact monotonically meta-Lindelöf T₂-space is first countable. In the last part of the note, we point out that there is a gap in Proposition 3.8 which appears in [H.R. Bennett, K.P. Hart, D.J. Lutzer, A note on monotonically metacompact spaces, Topology Appl. 157 (2) (2010) 456-465]. We finally give a detailed proof of how to overcome the gap.     

Selections for paraconvex-valued mappings on non-paracompact domains

We prove that Michael?s paraconvex-valued selection theorem for paracompact spaces remains true for C^′(E)-valued mappings defined on collectionwise normal spaces. Some possible generalisations are also given.     

Cover quasi-uniformities in frames

Quasi-uniformities (not necessarily symmetric uniformities) are usually studied via entourages (special neighbourhoods of the diagonal in X×X) where one can simply forget about the symmetry requirement. This has been done successfully in the point-free context as well, but there is a demand for a covering approach, a.o. because the point-free representation of the square X×X is not without difficulties. Based on the (spatial) ideas from Gantner and Steinlage (1972) [9], a cover type quasi-uniformity was developed in Frith (1987) [6] and other papers using biframes, the point-free variant of bitopologies. In this paper we show that this can be avoided and present a cover type quasi-uniformity structure enriching that of frame directly.     

Weakly continuously Urysohn spaces

We study weakly continuously Urysohn spaces, which were introduced in [P.L. Zenor, Continuously extending partial functions, Proc. Amer. Math. Soc. 135 (1) (2007) 305-312]. We show that every weakly continuously Urysohn wΔ-space has a base of countable order, that separable weakly continuously Urysohn spaces are submetrizable, hence continuously Urysohn, that monotonically normal weakly continuously Urysohn spaces are hereditarily paracompact, and that no linear extension of any uncountable subspace of the Sorgenfrey line is weakly continuously Urysohn. These results generalize various results in the literature concerning continuously Urysohn spaces.