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Abstract: For any space X, denote by the smallest (infinite) cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then , where denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality could be replaced by . Here we show that this can be done if X is also hereditarily normal. Moreover, we prove the following mapping theorem that involves the cardinal function . If is a continuous surjection of a countably compact space X onto a perfect space Y then . [Copyright &y& Elsevier]

Abstract: We prove that the Continuum Hypothesis is consistent with the statement that countably compact regular spaces that are hereditarily of countable π-character are either compact or contain an uncountable free sequence. As a corollary we solve a well-known open question by showing that the existence of a compact S-space of size greater than does not follow from the Continuum Hypothesis. [Copyright &y& Elsevier]

We give a finite combinatorial test for finite seminormal functors to possess the property O n and use it in establishing that in some cases this property leads to some well-known functors. For example, if some functor F possesses the property O 2 then F 2 coincides with either exp2 or the squaring functor. Hence we conclude that if F( D ω 1) and D ω 1 are homeomorphic then F 2 is either exp2 or (·)2. [ABSTRACT FROM AUTHOR]

Abstract: By a characterization of compact spaces in Section 1, a process of obtaining a compactification of an arbitrary topological space X is described in Section 2 by a combined approach of nets and open filters. The Wallman compactification can be embedded in if X is Hausdorff and by a little modification, the compactification of X is the Stone–Čech compactification of X if X is Tychonoff. [Copyright &y& Elsevier]

Given a space Y, let us say that a space X is a total extender for Y provided that every continuous map f :A→Y defined on a subspace A of X admits a continuous extension &ftilde; :X→Y over X. The first author and Alberto Marcone proved that a space X is hereditarily extremally disconnected and hereditarily normal if and only if it is a total extender for every compact metrizable space Y, and asked whether the same result holds without any assumption of metrizability on Y. We demonstrate that a hereditarily extremally disconnected, hereditarily normal, non-collectionwise Hausdorff space X constructed by Kenneth Kunen is not a total extender for K, the one-point compactification of the discrete space of size ω1. Under the assumption 2ω0=2ω1, we provide an example of a separable, hereditarily extremally disconnected, hereditarily normal space X that is not a total extender for K. Furthermore, using forcing we prove that, in the generic extension of a model of ZFC+MA(ω1), every first-countable separable space X of size ω1 has a finer topology τ on X such that (X,τ) is still separable and fails to be a total extender for K. We also show that a hereditarily extremally disconnected, hereditarily separable space X satisfying some stronger form of hereditary normality (so-called structural normality) is a total extender for every compact Hausdorff space, and we give a non-trivial example of such an X. [Copyright &y& Elsevier]

In this paper, we study the properties of 1-comparability and stable range one condition on ideals of regular rings, and we use these results to investigate normalized pseudo-rank functions on ideals and N*-complete ideals. These will generalize the corresponding results of Goodearl. Finally, we give some sufficient conditions under which P(I) is compact. [ABSTRACT FROM AUTHOR]

Some problems arising out of recent work of P.M. Gartside and J.T.H. Lo are here surveyed. [Copyright &y& Elsevier]

We study a new cardinal-valued invariant $ndw(X)$ (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)≤ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points. [ABSTRACT FROM AUTHOR]

We give characterizations for a subspace of a hyperspace, endowed with either the Vietoris or the Wijsman topology, to be compact or relatively compact. Then we characterize—both globally and at the single points—the local compactness of a hyperspace, endowed with either the Vietoris, the Wijsman, or the Hausdorff metric topology. Several examples illustrating the behaviour of local compactness are also included. [Copyright &y& Elsevier]

We prove that there exists a model of ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) in which there is a compact, metrizable, non-second countable, Cantor cube. This answers in the affirmative an open question by E. Wajch (2018) [15]. Furthermore, we strengthen a result in the above paper, namely "Countable products of metrizable spaces are quasi-metrizable implies van Douwen's Choice Principle", by first observing that the above topological statement implies the weak choice principle MC (ℵ 0 , ℵ 0) , i.e. "Every countably infinite set of countably infinite sets has a multiple choice function", and then by establishing that van Douwen's choice principle is strictly weaker than MC (ℵ 0 , ℵ 0) in ZF. The above independence result also resolves an open problem in P. Howard and J.E. Rubin (1998) [7] , and strengthens a result by P. Howard, K. Keremedis, H. Rubin, and J.E. Rubin (1998) [6]. [ABSTRACT FROM AUTHOR]

Abstract: For a compact space K we consider the set of all -points in K, i.e., points with countable base of neighborhoods in K. We show that, for every scattered Eberlein compact space K, the set is a -set in K. We also give an example of a scattered Eberlein compactum with non-metrizable set . Moreover, we give an example of a Corson compact space K such that does not contain any dense -subset of K. This answers three questions of Tkachuk. [Copyright &y& Elsevier]

Abstract: We prove the following result: If in a compact space X there is a λ-branching family of closed sets then X cannot be covered by fewer than λ many discrete subspaces. (A family of sets is λ-branching iff but one can form λ many pairwise disjoint intersections of subfamilies of .) The proof is based on a recent, still unpublished, lemma of G. Gruenhage. As a consequence, we obtain the following strengthening of the well-known Čech–Pospišil theorem: If X a is compact space such that all points have character then X cannot be covered by fewer than many discrete subspaces. [Copyright &y& Elsevier]

In set theory without the Axiom of Choice (AC), we investigate the deductive strength of the statements " every infinite Hausdorff space has a countably infinite cellular family " and " every infinite Hausdorff space has a countably infinite relatively discrete subspace ", and of variants of the above statements for certain classes of Hausdorff spaces, their mutual relationship, as well as their relationship with various weak choice principles. Among several results, we construct a model N of ZF (i.e., of Zermelo–Fraenkel set theory without AC) in which there exists a dense-in-itself, zero-dimensional, Hausdorff topology on ω such that c (ω) = s (ω) = ω (where c (ω) and s (ω) are the cellularity and the spread of ω , respectively), but ω has no infinite discrete subsets in N , and thus neither has infinite cellular families in N. [ABSTRACT FROM AUTHOR]

In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength (in terms of weak choice principles) of the following statements: "every infinite Hausdorff space has a denumerable cellular family", "every infinite Hausdorff space has a denumerable discrete subset", "every denumerable compact Hausdorff space has an infinite cellular family", and "every denumerable compact Hausdorff space has an infinite discrete subset", and answer open questions by E. Tachtsis "Infinite Hausdorff spaces may lack cellular families or discrete subsets of cardinality ℵ 0 " and an open question by A. Miller "Some interesting problems", which also appears in the former paper. [ABSTRACT FROM AUTHOR]