THE WALLMAN COMPACTIFICATION OF FRAMES

Abstract

We show that a B-conjunctive frame L, where B is a normal base for L gives rise to a strong inclusion on L and therefore a compactification of L. The resulting compact regular frame corresponds to the quotient frame obtained by Johnstone in his construction of the Wallman compactification for frames. It is also shown that, in the presence of pseudocompactness the Wallman compactification and the Wallman realcompactification coincide.     

More generalizations of pseudocompactness

We introduce a covering notion depending on two cardinals, which we call O-[μ,λ]-compactness, and which encompasses both pseudocompactness and many other known generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to O-[ω,ω]-compactness.We provide several characterizations of O-[μ,λ]-compactness, and we discuss its connection with D-pseudocompactness, for D an ultrafilter. The connection turns out to be rather strict when the above notions are considered with respect to products. In passing, we provide some conditions equivalent to D-pseudocompactness.Finally, we show that our methods provide a unified treatment both for O-[μ,λ]-compactness and for [μ,λ]-compactness.     

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.     

A note onR 1-topological spaces

The purpose of the present note is to give a number of characterizations of theR ₁-axiom and to show that theR ₁-axiom is equivalent to the weakly Hausdorff axiom introduced byB. Banaschewski andJ. M. Maranda [2]. In anR ₁-space it is shown that the locally compactness property is also open hereditary and that the closure of an almost compact set is the union of the closures of its points. A necessary and sufficient condition is obtained under which a locally compact set dense in anR ₁-space is open. Finally a variant of a well-known theorem regarding two continuous functions of a topological space into aT ₂-space is formulated forR ₁-spaces.     

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.     

Some aspects of (non) functoriality of natural discrete covers of locales

Abstract

The frame S_c(L) generated by closed sublocales of a locale L is known to be a natural Boolean (“discrete”) extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and S_c(L) are isomorphic. The construction S_c is not functorial; this leads to the question of individual liftings of homomorphisms L → M to homomorphisms S_c(L) → S_c(M). This is trivial for Boolean L and easy for a wide class of spatial L, M . Then, we show that one can lift all h : L → 2 for weakly Hausdor? L (and hence the spectra of L and S_c(L) are naturally isomorphic), and finally present liftings of h : L → M for regular L and arbitrary Boolean M.     

Ideals associated with realcompactness in pointfree function rings

Abstract

Let RL denote the ring of continuous real-valued functions on a com- pletely regular frame L. The support of an α ∈ RL is the closed quotient ↑(coz α)?. We show that if supports are coz-quotients in L, then the set of functions with realcompact support is an ideal. If L satisfies the stronger condition that supports are C-quotients, then this ideal is the intersection of pure parts of the free maximal ideals of RL. The set of functions whose cozeroes are realcompact is always an ideal, which is free if and only if L is locally realcompact if and only if L is (isomorphic to) an open quotient of υL. Further, this ideal is prime if and only if it is a free real maximal ideal if and only if υL → L is a one-point extension of L.     

STRONGLY SEQUENTIALLY CONTINUOUS FUNCTIONS

Abstract

Given a topological abelian group G, we study the class of strongly sequentially continuous functions on G. Strong sequential continuity is a property intermediate between sequential continuity and uniform sequential continuity, which appeared naturally in the study of smooth functions on Banach spaces. In this paper, we shall mainly concentrate on the gap between strong sequential continuity and uniform sequential continuity. It turns out that if G has some completeness property—for example, if it is completely metrizable—then all strongly sequentially continuous functions on G are uniformly sequentially continuous. On the other hand, we exhibit a large and natural class of groups for which the two notions differ. This class is defined by a property reminiscent of the classical Dirichlet theorem; it includes all dense sugroups of R generated by an increasing sequence of Dirichlet sets, and groups of the form (X, w), where X is a separable Banach space failing the Schur property. Finally, we show that the family of bounded, real-valued strongly sequentially continuous functions on G is a closed subalgebra of l∞(G).     

Some localized separation axioms and their implications

In a topological spaceX, a T₂-distinct pointx means that for anyyX xy, there exist disjoint open neighbourhoods ofx andy. Similarly, T₀-distinct points and T₁distinct points are defined. In a T_i-distinct point-setA, we assume that eachxA is a T _i -distinct point (i=0, 1, 2). In the present paper some implications of these notions which localize the T _i -separation axioms (i=0, 1, 2) requirement, are studied. Suitable variants of regularity and normality in terms of T₂-distinct points are shown hold in a paracompact space (without the assumption of any separation axioms). Later T₀-distinct points are used to give two characterizations of the R _D -axiom.¹ In the end, some simple results are presented including a condition under which an almost compact set is closed and a result regarding two continuous functions from a topological space into a Hausdorff space is sharpened. A result which relates a limit pointv to an -limit point is stated.     

RELATIVE COMPACTNESS AND COMPACTNESS OF GENERAL SUBSETS IN AN I- TOPOLOGICAL SPACE

Abstract

In a previous work, we introduced a form of compactness applicable to general fuzzy sets in an I-topological space. It was shown that many of the standard results for compactness in general topology remain valid in the fuzzy setting. In this paper we continue our investigations into the behaviour of compact fuzzy subsets. We also introduce the notion of a relatively compact fuzzy subset and obtain results very similar to those of general topology. Many of our results are in the setting of fuzzy neighbourhood space and fuzzy uniform spaces. In particular, a number of criteria for compactness, already known for the whole space, are extended to arbitrary fuzzy subsets in a fuzzy neighbourhood space.     

On the Urysohn number of a topological space II

Abstract

Some variations of Arhangel'skii inequality ∣X∣ = 2^χ(X)L(X) for every Hausdorff space X [3], given in [2] and [6] are improved.     

Completely prime L-filters,irreducible L-filters and sobriety

In this paper, for a frame L, we characterize modified sobriety in stratified L-topological spaces and strong L-topological spaces internally using certain classes of stratified L-filters. While the first characterization using completely prime L-filters is trivial and applies to all stratified L-topological spaces, the other one using irreducible L-filters generalizes an approach of R.E. Ho?mann to the lattice-valued case but is restricted to either the case that the lattice is a complete Boolean algebra or to the case of completely distributive lattices and strong L-topological spaces.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

On Pseudocompact, Countably Compact, Locally Bicompact Mappings, and k-Mappings

We consider various definitions of a pseudocompact mapping and the basic properties of pseudocompact mappings. Moreover, we consider the definition of countable compactness of a continuous mapping and study the properties of a countably compact mapping similar to the corresponding properties for countably compact spaces and also the interrelation between countable compactness and pseudocompactness of mappings. We also extend the notions of local bicompactness and k-space to continuous mappings.     

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     

On a problem concerning generalizations of realcompact spaces

PF-compact spaces are defined. Every almost realcompact space is PF-compact and every separable PF-compact space is almost realcompact. Nonetheless in the constructible universe V = L, a space of cardinality ?₁ is PF-compact if, and only if, the closure of each of its countable subsets is almost realcompact, and ω₁ is hereditarily PF-compact.     

On cardinality bounds involving the weak Lindelöf degree

We give a general closing-off argument in Theorem 2.3 from which several corollaries follow, including (1) if X is a locally compact Hausdorff space then |X| ≤ 2^wL(X)ψ(X), and (2) if X is a locally compact power homogeneous Hausdorff space then |X| ≤ 2^wL(X)t(X). The first extends the well-known cardinality bound 2^ψ(X) for a compactum X in a new direction. As |X| ≤ 2^wL(X)χ(X) for a normal spaceX[4], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.12 we give a short, direct proof of (1) that does not use 2.3. Yet 2.3 is broad enough to establish results much more general than (1), such as if X is a regular space with a π-base ? such that |B| ≤ 2^wL(X)χ(X) for all B ∈ ?, then |X| ≤ 2^wL(X)χ(X).

Separately, it is shown that if X is a regular space with a π-base whose elements have compact closure, then |X| ≤ 2^{wL(X)ψ(X)t(X)}. This partially answers a question from [4] and gives a third, separate proof of (1). We also show that if X is a weakly Lindelöf, normal, sequential space with χ(X) ≤ 2^?0, then |X| ≤ 2^?0.

Result (2) above is a new generalization of the cardinality bound 2^t(X) for a power homogeneous compactum X (Arhangel'skii, van Mill, and Ridderbos [3], De la Vega in the homogeneous case [10]). To this end we show that if U ? clD ? X, where X is power homogeneous and U is open, then |U| ≤ |D|^πχ(X). This is a strengthening of a result of Ridderbos [19].     

Congruences on ockham algebras with pseudocomplementation

The variety pOconsists of those algebras (L;?,?,f,^*,0,1) where (L;?,?,f,0,1) is an Ockham algebra, (L;?,?,f,^*,0,1) is a p-algebra, and the unary operations fand ^*. commute. For an algebra in pK _ωwe show that the compact congruences form a dual Stone lattice and use this to determine necessary and sufficient conditions for a principal congruence to be complemented. We also describe the lattice of subvarieties of pK _1,1identifying therein the biggest subvariety in which every principal congruence is complemented, and the biggest subvariety in which the intersection of two principal congruences is principal.     

Characterization ofL p (R n ) using Gabor frames

We characterize L^p norms of functions onR ⁿ for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L^p functions converge to the functions almost everywhere and in L^p for 1<p<∞. In L¹ we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L¹ in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L^p (R ⁿ) when 1≤p<∞.     

ALMOST 2-FULLY NORMAL,PAIRWISE PARACOMPACT AND COMPLETE DEVELOPABLE BISPACES

Abstract

Dedicated to Professor Sergio Salbany on the occasion of his 60th birthday.

We introduce and study the notion of an almost 2-fully normal bispace. In particular, we prove that a bispace is quasi-pseudometrizable if and only if it is almost 2-fully normal and pairwise developable. We obtain conditions under which an almost 2-fully normal bispace is subquasi-metrizable and show that the fine quasi-uniformity of any subquasi-metrizable topological space is bicomplete. We prove that every pairwise paracompact bispace (in the sense of Romaguera and Marin, 1988) is almost 2-fully normal and that the finest quasi-uniformity of any 2-Hausdorff pairwise paracompact bispace is bicomplete. We also characterize pairwise paracompactness in terms of a property of σ-Lebesgue type of the finest quasi-uniformity. Finally, we use Salbany's compactification of pairwise Tychonoff bispaces to characterize those bispaces that admit a bicomplete pair development and deduce that an interesting example of R. Fox of a non-quasi-metrizable pairwise stratifiable pairwise developable bispace admits a bicomplete pair development.