SMYTH COMPLETENESS IN TERMS OF NETS: THE GENERAL CASE

Abstract

Smyth completeness is the appropriate notion of completeness for quasi-uniform spaces carrying an additional topology to serve as domains of computation [2, 3]. The goal of this paper is to provide a better understanding of Smyth completeness by giving a characterization in terms of nets. We develop the notion of computational Cauchy net and an appropriate notion of strong convergence to get the result that a space is Smyth complete if and only if every computational Cauchy net strongly converges. As we are dealing with typically non-symmetric spaces, this is not an instance of the classical net-filter translation in general topology.     

Some new characterizations of pointfree pseudocompactness

Abstract

By [3], a frame L is pseudocompact iff every ??-sequence in L joining to the top terminates. Here it is shown, for any completely regular L, that pseudocompactness is also equivalent to (i) the analogous condition for ?-sequences, (ii) the countable almost compactness of L, (iii) the almost compactness of CozL as a σ-frame and (iv) the condition that every countably based proper filter in L clusters. Further we establish the zero-dimensional counterparts of the above, concerning the integer valued notion of pseudocompactness. Finally, we add to this a characterization of pseudocompactness in terms of uniformities.     

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.

     

When separability of the space of ultra-extremal functions is preserved

Abstract

The ultrametrically injective hull T_X of an ultrametric space (X, d) is investigated by viewing it as the space of ultra-extremal functions over X. It turns out that the ultra-extremal functions are also ultra-Ka?etov functions, satisfying two inequalities derived from the strong triangle inequality. We shall compare the ultra-extremal functions with some classes of functions defined with the help of one of the two inequalities from the definition of ultra-Kat?tov functions. We shall consider the question of when separability of the space of ultra-extremal functions is preserved.     

Duality in Function Spaces

In this paper we use Bartle’s technique to study duality between a topological space and a function space. Normally such a duality forms an essential part of Functional Analysis. We introduce several new topologies such as the topology of even convergence T_e, the closed-cocompact topology T_{k ′}, the (strong) local proximal convergence. We explore the topological groups of self-homeomorphisms of a topological space and shed light on the earlier work of Arens, Dieudonné, Di Concilio. We also study the concepts such as evenly equidistant, functionally equicontinuous, due to Bouziad-Troallic and topologically equicontinuous due to Royden. In memory of Professor Enrico Meccariello who made a considerable contribution to this work and who suddenly passed away before his time     

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 epireflective and coreflective hulls of a particular L-topological space

In this paper, we study some aspects of the category L-ZTop of zero-dimensional L-topological spaces. After noting that it is a topological category, we identify a ‘Sierpinski object’ L_Z in it. We further show that two epireflective hulls of L_Z respectively turn out to be the categories of zero-dimensional T₀-L-topological spaces and of zero-dimensional sober L-topological spaces. We also determine the coreflective hull of L_Z in the category of L-topological spaces.     

Comparison of Boolean algebras

In this work, some results related to superatomic Boolean interval algebras are presented, and proved in a topological way. Let x be an uncountable cardinal. To each I x, we can associate a superatomic interval Boolean algebra B _I of cardinality x in such a way that the following properties are equivalent: (i) I I x, (ii) B _I is a quotient algebra of B _J, and (iii) there is an homomorphism f from B _J into B _I such that for every atom b of B _I, there is an atom a of B _J satisfying f(a)=b. As a corollary, there are 2 ^x isomorphism types of superatomic interval Boolean algebras of cardinality x. This case is quite different from the countable one.     

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.     

Weaker forms of the Menger property in bitopological spaces

Abstract

In this paper we continue previous investigations on the weaker forms of the Menger property in bitopological spaces. We introduce weakly Menger property and study some topological properties of almost and weakly Menger bitopological spaces. We also consider the almost Hurewicz spaces in a bitopological context.     

On the division by R n

Summary In this paper we prove that, ifS ×R ⁿ is homeomorphic toR ^{n + 1}, thenS is homeomorphic toR.     

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.     

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).     

EMBEDDINGS INTO THE CATEGORY OF SUPER UNIFORM SPACES

Abstract

The category US of uniform spaces has been generalised in various ways. The category FUS, of fuzzy uniform spaces and the category GUS, of generalised uniform spaces have both been shown to be good extensions in the sense that US can be embedded into them. We show here that the category SUS, of super uniform spaces also enjoys this property and furthermore, the categories FUS and GUS can be embedded into SUS.     

PRODUCTS IN TOPOLOGY

Abstract

Examples are provided which demonstrate that in many cases topological products do not behave as they should. A new product for topological spaces is defined in a natural way by means of interior covers. In general this is no longer a topological space but can be interpreted as categorical product in a category larger than Top. For compact spaces the new product coincides with the old. There is a converse: For symmetric topological spaces X the following conditions are equivalent: (1) X is compact; (2) for each cardinal k the old and the new product X^k coincide; (3) for each compact Hausdorff space Y the old and the new product X x Y coincide. The new product preserves paracompactness, zero-dimensionality (in the covering sense), the Lindelöf property, and regular-closedness. With respect to the new product, a space is N-complete iff it is zerodimensional and R-complete.     

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

Lattice and Algebra Homomorphisms on C(X) in Zermelo-Fraenkel Set Theory

Abstract

Let C (X) denote the lattice-ordered algebra of all real-valued continuous functions on a topological space X. This paper discusses in Zermelo-Fraenkel Set Theory the equivalence on C (X) between algebra homomorphisms, lattice homo- morphisms, and point evaluations.     

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].     

A New Characterization of the Continuous Functions on a Locale

Within the category W of archimedean lattice-ordered groups with weak order unit, we show that the objects of the form C(L), the set of continuous real-valued functions on a locale L, are precisely those which are divisible and complete with respect to a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of a W-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as c³A, the ``closed under countable composition hull of A,' as C(Y_lA), where Y_lA is the Yosida locale of A, and as the largest essential reflection of A.     

HAUSDORFF NEARNESS SPACES

Abstract

A categorical characterization of the category Haus of Hausdorft topological spaces within the category Top of topological spaces is given. A notion of a Hausdorff nearness space is then introduced and it is proved that the resulting subcategory Haus Near of the category Near of nearness spaces fulfills exactly the same characterization as derived for Haus in Top. Properties of Haus Near and relations to other important sub-categories of Near are studied.