Weak covering properties and selection principles

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster?s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most _ℵ1${\aleph }_1$. It turns out that topological spaces having Alster?s property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster?s property satisfy a seemingly stronger version of Alster?s property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.     

Nice separation axioms

The epi-reflective hull B of an objectB in the category of locales is called anice separation axiom provided it enjoys certain properties which are natural generalizations of properties satisfied by the categories of completely regular locales and zero dimensional locales. Invariably, a family ofStone-like duality theorems ensues, distinguishing (for each large enough regular cardinal ) the full subcategory of all -LindelöfB-objects. Some corollaries for topological spaces arise, as well as some open problems, upon taking of spatial parts.This paper, consisting primarily of excerpts from the author's Ph.D. thesis, Wesleyan University, Middletown, Connecticut (1992), also represents, in part, the results of research engaged in at Charles University, Prague, during the 1992–1993 academic year, with the support of the United States Information Agency and the Fulbright Program.     

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.     

Hereditarily strongly cwH and other separation axioms

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ₁), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD(ℵ₁) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω₁.     

Hereditary normality-type properties of hyperspaces

Let X be a Hausdorff topological space and exp(X) be the space of all (nonempty) closed subsets of a space X with the Vietoris topology. We consider hereditary normality-type properties of exp(X). In particular, we prove that if exp(X) is hereditarily D-normal, then X is a metrizable compact space.     

Expandable network and covering properties for uniform Eberlein compacta

In this paper we characterize the class of uniform Eberlein compact spaces through a network condition and also in terms of covering properties for the square of the space.     

Interior operators and topological separation

A notion of separation with respect to an interior operator in topology is introduced and some basic properties are presented. In particular, it is shown that this notion of separation with respect to an interior operator gives rise to a Galois connection between the collection of all subclasses of the class of topological spaces and the collection of all interior operators in topology. Characterizations of the fixed points of this Galois connection are given and examples are provided.     

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.     

Remarks on countability and star covering properties

In this paper, we construct an example of a T₄ feebly Lindelöf space X which is not star Lindelöf under ℵ₀²=ℵ₁², which gives a partial answer to Alas, Junqueira and Wilson (2011) [1, Question 4].     

Weak regularity and consecutive topologizations and regularizations of pretopologies

L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ-compact pretopology. On the other hand, it is proved that for each n<ω there is a (regular) pretopology ρ (on a set of cardinality c) such that k(RT)ρ>n(RT)ρ for each k<n and n(RT)ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT-order ?ω₀. Moreover, all these pretopologies have the property that all the points except one are topological and regular.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

QN-spaces, wQN-spaces and covering properties

The main results of the paper are as follows: covering characterizations of wQN-spaces, covering characterizations of QN-spaces and a theorem saying that Cp(X) has the Arkhangel'ski?ˇ property (α₁) provided that X is a QN-space. The latter statement solves a problem posed by M. Scheepers [M. Scheepers, Cp(X) and Arhangel'ski?ˇ's αi-spaces, Topology Appl. 89 (1998) 265-275] and for Tychonoff spaces was independently proved by M. Sakai [M. Sakai, The sequence selection properties of Cp(X), Preprint, April 25, 2006]. As the most interesting result we consider the equivalence that a normal topological space X is a wQN-space if and only if X has the property S₁(Γshr,Γ). Moreover we show that X is a QN-space if and only if Cp(X) has the property (α₀), and for perfectly normal spaces, if and only if X has the covering property (β₃).     

Separation axioms and frame representation of some topological facts

Similarly as the sobriety is essential for representing continuous maps as frame homo-morphisms, also other separation axioms play a basic role in expressing topological phenomena in frame language. In particular,T _D is equivalent with the correctness of viewing subspaces as sublocates, or with representability of open or closed maps as open or closed homomorphisms. A weaker separation axiom is equivalent with an algebraic recognizability whether the intersection of a system of open sets remains open or not. The role of sobriety is also being analyzed in some detail.In honour of Nico Pumplün on the occasion of his 60th birthdayThe support of the Italian C.N.R. is gratefully acknowledged.Partial financial support of the Italian M.U.R.S.T. is gratefully acknowledged.     

Products of monotonically normal spaces with factors defined by topological games

Recently, it has been proved that orthocompactness implies normality for the products of a monotonically normal space and a compact space. It had been known that normality, collectionwise normality and the shrinking property are equivalent for the same products. We extend these two results for the products replacing the compact factor with a factor defined by topological games. Moreover, we prove the equivalence of orthocompactness and weak suborthocompactness in these products.     

Internal normality and internal compactness

This text contains an example which presents a way to modify any Dowker space to get a normal space X such that X×[0,1] is not κ-normal, and a theorem implying the existence of a non-Tychonoff space which is internally compact in a larger regular space. It gives answers to several questions by Arhangel'skii [A.V. Arhangel'skii, Relative normality and dense subspaces, Topology Appl. 123 (2002) 27-36].     

Certain sequences with compact closure

This paper deals with a question which is stated by quite simple definitions. A sequence {xn} in a space X is called a β-sequence if every subsequence of it has a cluster point in X. The closure of the sequence {xn} means the closure of in X. Here we consider the question when a β-sequence has compact closure. We give several answers to this question.     

Hausdorff separation in categories

Considering subobjects, points and a closure operator in an abstract category, we introduce a generalization of the Hausdorff separation axiom for topological spaces: the notion ofT ₂-object. We discuss the properties ofT ₂-objects, which depend essentially on the behaviour of points, and finally we relate them to the well-known separated objects.The results of this paper are essentially taken from the author's Ph. D. Thesis written under the supervision of Professors M. Sobral and W. Tholen and partially supported by a scholarship of I.N.I.C.-Instituto Nacional de Investigação Científica.