H-closed Spaces Research Articles

Three small results on normal first countable linearly H-closed spaces

Three small results on normal first countable linearly H-closed spaces

Linearly S-closed spaces

We introduce the class of linearly S-closed spaces as a proper subclass of linearly H-closed spaces. This property lies between S-closedness and countable S-closedness. A space is called linearly S-closed if and only if any semi-open chain cover posses a member dense in the space. It is shown that in the class of extremally disconnected spaces the class of linearly H-closed spaces and linearly S-closed spaces coincide. We gave characterizations of these spaces in terms of s-accumulation points of chain filter bases and complete s-accumulation points of families of open subsets. While regular S-closed spaces are compact there is a non compact, regular, linearly S-closed space. It is shown that a Hausdorff, first countable, linearly S-closed space is extremally disconnected. Moreover, in the class of first countable, regular, compact spaces the notions of S-closedness, linearly S-closedness and extremally disconnectedness are equivalent. Some cardinality bounds for this class of spaces are obtained. Several examples are provided to illustrate our results.

Locally H-closed Spaces, Subspaces and Their Extensions

The primary goal is to characterize Locally H-closed spaces (LHC), by conditions on the remainders of their extensions. These spaces are also characterized using subspaces and their extensions as well. Characterizing these classes of spaces using the remainders of the subspaces in their extensions provide characterizations of them in terms of their boundaries. Recently, the authors have proved that these results give necessary and sufficient conditions for the space to be compact A number of equivalences are proved for Hausdorff (Urysohn) [regular] spaces. These results lead to similar characterizations of Locally Urysohn-closed (LUC) as well as Locally regular-closed (LRC) spaces. Some of these equivalent properties generalize a number of existing results on these topics. In the present article it is shown that if X is a Hausdorff LHC space then each closed set is an intersection of regularly open sets as well as each closed set is an intersection of semi-closed neighborhoods. In 1969 Porter and Thomas had shown that in a Hausdorff space a locally H-closed subspace is the intersection of an open set and a closed set. In this article, it is shown that a space X is LHC if and only if every nonempty proper regularly closed subset of X is LHC.

A common extension of Lindelöf, H-closed and ccc

The inequality |X|≤2χ(X) has been proved to be true for Lindelöf spaces (Arhangel'skiĭ, 1969), H-closed spaces (Dow-Porter, 1982) and ccc spaces (Hajnal-Juhász 1967), by quite different arguments. We present a common extension of all these properties which allows us to give a unified proof of these three theorems.

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.

A useful common weakening of Lindelöf and H-closed

The inequality |X|≤2χ(X) has been proved to be true for both the class of Lindelöf spaces (Arhangel'skii, 1969 [1]) and that of H-closed spaces (Dow and Porter, 1982 [6]), by different arguments. We present a common weakening of the Lindelöf and H-closed properties which allows us to give a unified proof of these two theorems.

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.

On the cardinality of Hausdorff spaces and H-closed spaces

We introduce the cardinal invariant aL′(X) and show that |X|≤2aL′(X)χ(X) for any Hausdorff space X (a corollary of Theorem 4.4). This invariant has the properties a) aL′(X)=ℵ0 if X is H-closed, and b) aL(X)≤aL′(X)≤aLc(X). Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound 2L(X)χ(X) from which it follows that |X|≤2ψc(X) if X is H-closed (Dow/Porter [5]). The invariant aL′(X) is constructed using convergent open ultrafilters and an operator cˆ:P(X)→P(X) with the property clA⊆cˆ(A)⊆clθ(A) for all A⊆X. As a comparison with this open ultrafilter approach, in §3 we additionally give a κ-filter variation of Hodel's proof [8] of the Dow–Porter result. Finally, for an infinite cardinal κ, in §5 we introduce κwH-closed spaces, κH′-closed spaces, and κH″-closed spaces. The first two notions generalize the H-closed property. Key results in this connection are that a) if κ is an infinite cardinal and X a κwH-closed space with a dense set of isolated points such that χ(X)≤κ, then |X|≤2κ, and b) if X is κH′-closed or κH″-closed then aL′(X)≤κ. This latter result relates these notions to the invariant aL′(X) and the operator cˆ.

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.

Local fuzzy compactness with respect to fuzzy ideal

Given a nonempty set X, a fuzzy ideal I on X is a collection of subsets of X closed under finite union and subset operations. In this paper we define a space to be locally I fuzzy compact space if every point in the space has an I fuzzy compact neighbourhood. Basic results concerning locally I fuzzy compact spaces are given relating to subspaces, preservation of functions, and products. Classical results concerning locally fuzzy compact spaces are obtained by I ={φ},and certain results for locally H-closed spaces are obtained by letting I be the fuzzy ideal of nowhere dense sets. Locally H-closed spaces are characterized in the category of Hausdorff spaces as being the locally nowhere-dense-fuzzy compact sets.

Nearly H-closed spaces

For a subset M of a topological space X, the θ-closure {ie626-01} M is defined as the set of all x ∈ X such that any closed neighborhood of x intersects M. A Urysohn space X is said to be U-closed if, whenever X ∪ {ξ} is a Urysohn space obtained from X by adding one point ξ, the point ξ is isolated in X ∪ {ξ}. The θ-closure operator is applied to study compactness-type properties of (weakly) U-and H-closed and closed-hereditarily U-and H-closed spaces.

Minimal sequential Hausdorff spaces

A sequential space (X, T) is called minimal sequential if no sequential topology on X is strictly weaker than T. This paper begins the study of minimal sequential Hausdorff spaces. Characterizations of minimal sequential Hausdorff spaces are obtained using filter bases, sequences, and functions satisfying certain graph conditions. Relationships between this class of spaces and other classes of spaces, for example, minimal Hausdorff spaces, countably compact spaces, H‐closed spaces, SQ‐closed spaces, and subspaces of minimal sequential spaces, are investigated. While the property of being sequential is not (in general) preserved by products, some information is provided on the question of when the product of minimal sequential spaces is minimal sequential.

Embeddings in the upper Stone-Čech compactification

In this paper a seventeen year old question by Porter is answered showing that the Banaschewski-Fomin-Šanin extension μX of a space X can be embedded in the upper Stone-Čech compactification, β + X. Frolik and Liu characterized H-closed spaces as maximal Hausdorff subspaces in their closure in Π C+( X) I +; that is, X is H-closed iff e[ X] is a maximal Hausdorff subspace of β + X. This result is strengthened/generalized by showing that σX, μX and, in fact, every strict H-closed extension of X can be embedded in β + X. However, there may be infinitely many copies of every strict H-closed extension within β + X.

Properties of Locally H-Closed Spaces

This paper investigates the properties of locally $H$-closed spaces with regard to extensions, subspaces, and functions. We solve the $H$-closed extension remainder problem by showing that a space is locally $H$-closed if and only if it has a $\theta$-closed remainder in some $H$-closed extension. In fact, an $H$-closed space is Urysohn iff every $H$-closed subspace is $\theta$-closed. We solve the locally $H$-closed subspace problem by giving a necessary and sufficient condition for a subspace of a locally $H$-closed space to be locally $H$-closed. In particular, an open subspace of a locally $H$-closed space is locally $H$-closed if and only if its boundary is a $\theta$-closed subspace of its closure. An $H$-closed space is shown to be compact if and only if every open subset is locally $H$-closed. A retract of a locally $H$-closed space is locally $H$-closed.

Properties of locally H-closed spaces

This paper investigates the properties of locally H H -closed spaces with regard to extensions, subspaces, and functions. We solve the H H -closed extension remainder problem by showing that a space is locally H H -closed if and only if it has a θ \theta -closed remainder in some H H -closed extension. In fact, an H H -closed space is Urysohn iff every H H -closed subspace is θ \theta -closed. We solve the locally H H -closed subspace problem by giving a necessary and sufficient condition for a subspace of a locally H H -closed space to be locally H H -closed. In particular, an open subspace of a locally H H -closed space is locally H H -closed if and only if its boundary is a θ \theta -closed subspace of its closure. An H H -closed space is shown to be compact if and only if every open subset is locally H H -closed. A retract of a locally H H -closed space is locally H H -closed.

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.

Remainders of H-closed extensions

The remainders of compactifications of Tychonoff spaces have been studied extensively during the last five decades, but, the remainders of H-closed extensions of Hausdorff spaces have not been studied in a concerted manner so far. In this paper, we show that two Hausdorff spaces X and Y have the same remainders in their H-closed extensions if there exists a perfect, irreducible and θ-continuous mapping from X onto Y. Further, a Hausdorff space Z is shown to be the remainder of some H-closed extension of a given space X if and only if there exists a perfect (not necessarily continuous) mapping from σX⧹ X onto Z. We also show that if X is locally H-closed and |σX⧹X|⩾ℵ 0 , then the set of remainders of H-closed extensions of X contains all separable H-closed spaces. We give examples to illustrate the differences between the sets of remainders of compactifications and remainders of H-closed extensions of a fixed Tychonoff space.

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.

Accumulation Points of Nowhere Dense Sets in H-Closed Spaces

Accumulation Points of Nowhere Dense Sets in H-Closed Spaces

Epireflective subcategories of Hausdorff categories

Let pHaus denote the category of Hausdorff spaces and p-maps, and let HCL denote the subcategory of pHaus consisting of H-closed spaces and continuous functions. It is well-known that HCL is an epireflective subcategory of pHaus. In this paper we characterize the epireflective subcategories of pHaus that contain HCL.