H-closed Space Research Articles

Absolutes and n -H-closed spaces

In this paper the investigation of n -H-closed spaces that was started by Basile et al . (2019) is continued for every n ∈ ω, n ≥ 2. In particular, starting with the relationship between the absolute space PX of an arbitrary topological space X , reported by Ponomarev and Shapiro (1976) and introduced by Blaszczyk (1975, 1977), Ul'yanov (1975a,b) and Shapiro (1976), it is shown that the absolute PX is n -H-closed if and only if X is n -H-closed. For an arbitrary space X , a β-like extension (β for the Stone-Cech compactification) Y is constructed for the semiregularization PX(s) of the absolute PX such that Y is a compact, extremally disconnected, completely regular (but not necessarily Hausdorff) extension of PX(s) , and PX(s) is C* -embedded in Y . The definition of the Fomin extension σ X for a Hausdorff space X (Porter and Woods 1988) is extended to an arbitrary space X and σ X X is shown to be homeomorphic to the remainder Y PX(s) . A similar result is established when X is an n -Hausdorff space defined by Basile et al. (2019). Further, we give a cardinality bound for any n -Hausdorff space X and show that the inequality |X| ≤ 2^χ( X ) for an H-closed space X proved by Dow and Porter (1982) can be extended to n -H-closed spaces.

On linearly H-closed spaces

We study a subclass of the class of pseudocompact spaces, the linearly H-closed spaces, recently introduced by M. Baillif. We give two new characterizations of these spaces in terms of filters and complete accumulation points of families of open sets. We show that a monotonically normal linearly H-closed space is compact and that linear H-closedness is preserved under a product with an H-closed space. Additionally, a condition is given in order that the property be preserved under countable products.

Observations on some cardinality bounds

We try to find a common extension of two cardinal inequalities for Lindelöf spaces. Using an estimate of the number of Gδ points due to Balogh, we improve a result of Juhász and Spadaro. A cardinal inequality for linearly Lindelöf Tychonoff spaces proved by Arhangel'skiĭ and Buzyakova should be actually true for Hausdorff spaces. We observe this happens under some restrictions on cardinal arithmetics, including a consequence of Martin's axiom. Finally, we address the question to estimate the cardinality of a first countable linearly H-closed space.

Cardinal functions Fθ(X) and tθ(X) for H-closed spaces

This article extends the results proved by Cammaroto and Kočinac in 1993 by showing that tθ(X) = tθ(Xs) = t(Xs) = Fθ(X) = Fθ(Xs) = F(Xs) for every H-closed and Urysohn space. Examples are developed to study all of the relationships among the cardinal functions t, tθ, F and Fθ in the context of H-closed and Urysohn spaces. Also, an H-closed space H is constructed for which Fθ(H) < tθ(H). Two questions posed by Cammaroto and Kočinac about H-closed spaces are completely answered.

Separating H-sets by Open Sets

AbstractIn an H-closed, Urysohn space, disjoint H-sets can be separated by disjoint open sets. This is not true for an arbitrary H-closed space even if one of the H-sets is a point. In this paper, we provide a systematic study of those spaces in which disjoint H-sets can be separated by disjoint open sets.

Lower and upper topologies in the Hausdorff partial order on a fixed set

In the partial order of Hausdorff topologies on a fixed infinite set there may exist topologies τ ⊊ σ in which there is no Hausdorff topology μ satisfying σ ⊊ μ ⊊ τ . τ and σ are lower and upper topologies in this partial order, respectively. Alas and Wilson showed that a compact Hausdorff space cannot contain a maximal point and therefore its topology is not lower. We generalize this result by showing that a maximal point in an H-closed space is not a regular point. Furthermore, we construct in ZFC an example of a countably compact, countably tight lower topology, answering a question of Alas and Wilson. Finally, we characterize topologies that are upper in this partial order as simple extension topologies.

On Katětov Spaces

AbstractRecent work by Krystock, Porter, and Vermeer has emphasized the importance of the concepts of Katětov spaces and H-sets in the theory of H-closed spaces. These properties are closely related to being the θ-closure of some set and being the adherence of an open filter. This relationship is developed by establishing, among other facts, that an H-closed space in which every closed set is the θ-closure of some set is compact and the θ-closure of a subset of an H-closed space is Katětov and characterizing the open filter adhérences of a space as precisely those sets which are the image of a closed set of the absolute of the space. Also, examples are given of a countable, scattered space which is not Katětov and an H-closed space with an H-closed subspace which is not the θ-closure of any subset of the given space.

ON H-Sets and Open Filter Adherences(1)

AbstractThe relationship between H-sets and open filter adhérences is considered. The open filter adhérences of an H-closed space are shown to be H-sets; and, a necessary and sufficient condition is given for an H-set S, of a Hausdorff space X, to be an open filter adherence. A necessary condition is determined for the existence of a minimal adherent set which contains S; and, in the case that X is H-closed, sufficient conditions are determined. As a related result, an H-closed space X is shown to be seminormal if every H-set of X possesses a neighborhood base consisting of regular open sets.

Accumulation points of nowhere dense sets in $H$-closed spaces

In this paper a ten year old problem by Kulpa and Szymanski is settled by constructing an example of a minimal Hausdorff space without isolated points which has a point that is not the accumulation point of any nowhere dense subset of the minimal Hausdorff space. Also, a result by Kulpa and Szymanski is extended by showing that a regular point in an H-closed space without isolated points is the accumulation point of some nowhere dense subset. A decade ago, Kulpa and Szymafnski [KS] showed that each point in a compact Hausdorff space without isolated points is the accumulation point of a nowhere dense set and they asked if the result is true in the setting of minimal Hausdorff spaces without isolated points. They noted that the result is false for arbitrary H-closed spaces without isolated points. In this paper, we settle this question by giving an example of a minimal Hausdorff space without isolated points and a point which is not the accumulation point of any nowhere dense set. Also, we extend Kulpa and Szymanski's result by showing that a regular point in an H-closed space without isolated points is the accumulation point of some nowhere dense subset. First, a few preliminary results and definitions are needed. All spaces considered in this paper are assumed to be Hausdorff. A space X is H-closed if X is closed in every space in which X is a subspace. An open set U in a space X is said to be regular open if U = int(cl U). A space X is semiregular if { U c X: U is regular open} is a base for the topology of X and is minimal Hausdorff if there is no strictly coarser Hausdorff topology on the space. Katetov [K] has shown that a space is minimal Hausdorff iff it is H-closed and semiregular. For a space X, let tX = X U {q: qi is a free open ultrafilter on X}. For each open set U c X, let oU= U U { E It X\X: int(clV) c U for some Ve 9/}. THEOREM 1. Let X be a semiregular space. The following are true: (a) [B, F, S] The family {oU: U is regular open in X} is a base for a minimal Hausdorff extension ttX of X. (b) [PW] For a regular open subset U of X, cl,,XoU = oU U cl xU. D Received by the editors May 29, 1984. 1980 Mathematics Subject Classification. Primary 54D25.

Resolutions of 𝐻-closed spaces

It is shown that every Hausdorff space X is the perfect, irreducible, continuous image of a Hausdorff space X ~ \tilde X which has a basis with open closures. Further, w ( X ~ ) ⩽ w ( X ) w(\tilde X) \leqslant w(X) , where w ( X ~ ) w(\tilde X) represents the weight of X, and if X is H-closed then X ~ \tilde X is also H-closed. A corollary of this result is that if f : X → Y f:X \to Y is a continuous map of the H-closed space X onto the semi-regular Hausdorff space Y, then w ( Y ) ⩽ w ( X ) w(Y) \leqslant w(X) .

Inverse limits and absolutes of 𝐻-closed spaces

The inverse limit of an inverse system of nonempty (respectively, minimal Hausdorff) H-closed spaces with continuous open surjective bonding maps is shown to be a nonempty H-closed (respectively, minimal Hausdorff) space. This result is used to resolve a conjecture, in the affirmative, of S. W. Willard: Every H-closed space is the continuous and open image of a minimal Hausdorff space.

A Note on H-Closed Extensions of a Product

Any Hausdorff space X is a dense subspace of an H-closed space κX, called the Katětov extension of X, with the property that any H-closed extension Y of X is a continuous image of κX under a mapping which leaves X pointwise fixed [8], [10]. In [8], Liu has shown that the extensions κ(X×Y) and κX×κY of X × Y are equal iff (1) X or Y is finite, or (2) X and Y are H-closed. In this note, we examine whether homeomorphism of these two extensions implies equality. We give a condition under which homeomorphism implies equality and an example to show that this relation does not hold in general.