The properties of being cellular-w1-compact and maximal cellular-w1-compact and their relationships with certain star covering properties are studied. Some conditions which imply that a cellular-w1-compact space is w1-compact are obtained. Characterizations of the compact productivity of both cellular-w1-compactness and almost cellular-w1-compactness are given and it is shown that in the class of perfect spaces these latter two properties coincide. A number of properties of maximal cellular-w1-compact spaces are obtained and it is shown that if such a space is crowded, then it is a weak P-space. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We continue the study of cellular-compact spaces and the larger class of cellular-countably-compact spaces. We give a number of sufficient conditions involving local bases and local π -bases in order that a cellular-countably-compact space be countably compact and some conditions which imply that a topology is maximal with respect to being cellular-countably-compact are obtained. We also consider the compact productivity of the previously mentioned properties and give a characterization of those spaces whose product with a compact space is almost cellular-countably-compact.

A subclass of the class of feebly Lindelöf spaces, namely the almost cellular-Lindelöf spaces, which contains the class of all cellular-Lindelöf spaces is studied. A necessary and sufficient condition is obtained in order that the product of a Hausdorff space and a compact space be almost cellular-Lindelöf as well as a sufficient condition in order that an almost cellular-Lindelöf space be weakly Lindelöf. Additionally a cardinality bound for feebly Lindelöf spaces with a rank 3 diagonal is given and an example of a locally compact, cellular-compact space which is not weakly Lindelöf is given.

We establish that any Hausdorff discretely κ-Lindelöf space X must be κ-Lindelöf if t(X)≤κ. Besides, in κ-Lindelöf spaces whose tightness does not exceed κ, the property of having pseudocharacter ≤κ must be discretely reflexive. Under the hypothesis c<ωω we prove that countable pseudocharacter is discretely reflexive in Lindelöf Σ-spaces. If X is a Tychonoff space and ψ(D‾)≤κ for any discrete set D⊂Cp(X), then κ+ is a caliber of X, we have the inequality hd(X×X)≤2κ and ψ(Y‾)≤κ for any Y∈[Cp(X)]≤κ+; this implies that ψ(Cp(X))≤2κ. We also show that the property of having pseudocharacter ≤κ is discretely reflexive in Cp(X) whenever X is a compact space with t(X)≤κ.

We study two subclasses of the class of feebly compact spaces, namely weakly cellular-compact spaces and almost cellular-compact spaces. The first of these was introduced and studied by Pichardo-Mendoza, Tamariz-Mascarua and Villegas-Rodriguez in the class of Tychonoff spaces, while the second is a subclass of the first which contains all cellular-compact spaces. We show that many of the results obtained by the above mentioned authors are valid in the class of Hausdorff spaces and we give conditions under which the product of an almost cellular-compact space and a compact space is almost cellular-compact. The main consistency results concern the preservation of this property under products with compact spaces.

We study three subclasses of the class of pseudocompact spaces. We answer two open questions concerning cellular-compact spaces and another concerning cellular-Lindelöf spaces and introduce and study three other subclasses of the class of feebly compact spaces, namely the class of cellular-countably-compact spaces, that of the cellular-sequentially-compact spaces and that of the cellular-compact-metrizable spaces. We show that it is independent of ZFC whether or not a cellular-compact-metrizable LOTS is metrizable.

A discrete subset is said to be Lindelöf dominated (respectively, $$\omega$$-dominated) if it is contained in the closure of a Lindelöf (respectively, a countable) subspace. We continue the study of spaces begun in [1] in which every discrete subset is Lindelöf dominated (respectively, $$\omega$$-dominated). We generalize results of [1] and [15] concerning perfect Hausdorff spaces and give a ZFC example of a perfect space in which all discrete subsets are Lindelöf dominated but not $$\omega$$-dominated.

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.

We establish that a first countable \(\omega \)-monolithic space is star countable if and only if it has countable extent. A consistent example is given of a first countable normal star Lindelöf space of uncountable extent. Under the continuum hypothesis we prove that for any compact K, the space \(C_p(K)\) is star countable if and only if it is Lindelöf. The above-mentioned results answer several published open questions.

We study two subclasses of the class of feebly compact spaces, namely those classes having the properties appearing in the title. Dorantes-Aldama and Shakhmatov in [6] have given a number of conditions equivalent to selective pseudocompactness in the class of Tychonoff spaces and we show here that these condtions are also equivalent, both in the class of Hausdorff spaces and also in a subclass of the class of T-1-spaces. We characterize both maximal selective feeble compactness and maximal sequential pseudocompactness in the class of T-1-spaces and consider the problem of when a Tychonoff space has a sequentially pseudocompact compactification.

Two star properties recently studied by Song and a generalization of countable compactness called weak star finiteness by Song and previously, 1-cl-starcompactness by Matveev and Ikenaga, are studied. We show that two of these properties coincide with feeble Lindelöfness and feeble compactness respectively in the class of spaces with a dense set of isolated points. Preservation of one of these properties under products by compact and sequentially compact spaces is also considered.

A topological property \(\mathscr {P}\) is reflected in continuous images of weight at most \(\omega _1\) if a space X has \(\mathscr {P}\) whenever every continuous image of X of weight at most \(\omega _1\) has \(\mathscr {P}\). When X is a generalized ordered space, we consider a number of topological properties including feeble Lindelöfness and \(\kappa \)-monolithicity. In the final section we study small images of pseudocompact and countably compact spaces; we give a condition on continuous images of a pseudocompact space in order that it be compact and show that it is consistently true that a countably compact space of countable projective tightness is countably tight.

A space X is discretely generated at a point x∈X if for any A⊆X with x∈cl(A), there exists a discrete set D⊆A such that x∈cl(D). The space X is discretely generated if it is discretely generated at every point x∈X. We say that X is weakly discretely generated if for any non-closed set A⊆X, there exists a discrete set D⊆A such that cl(D)\A≠∅. We obtain new results concerning these properties in the class of P-spaces.

We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.

A space X is discretely generated at a point \({x \in X}\) if for any \({A \subseteqq X}\) with \({x \in \textsf{cl}(A)}\) , there exists a discrete set \({D \subseteqq A}\) such that \({x \in \textsf{cl}(D)}\) . The space X is discretely generated if it is discretely generated at every point \({x \in X}\) . We say that X is weakly discretely generated if for any non-closed set \({A \subseteqq X}\) , there exists a discrete set \({D \subseteqq A}\) such that \({\textsf{cl}(D) \setminus A \neq \emptyset}\) . New results about these properties in the classes of pseudocompact and Čech-complete spaces are obtained and a theorem of Ivanov and Osipov concerning the ordinal function idc is generalized to the class of Čech-complete spaces.

We correct the proof of Theorem 2.9 of the paper mentioned in the title (published in Applied General Topology, 13 No.1 (2012), 11-19).

A space X is discretely generated at a point x∈X if for any A⊆X with x∈cl(A), there exists a discrete set D⊆A such that x∈cl(D). The space X is discretely generated if it is discretely generated at every point x∈X. We say that X is weakly discretely generated if for any non-closed set A⊆X, there exists a discrete set D⊆A such that cl(D)∖A≠∅. We obtain new results concerning products of spaces belonging to these classes.

In this paper a space X is pseudocompact if it is Tychonoff and every real-valued continuous function on X is bounded. We obtain conditions under which a Tychonoff space is maximal pseudocompact and study conditions under which a regular space is maximal R-closed.

Whenever P is a topological property, we say that a topological space is star P if whenever U is an open cover of X, there is a subspace A⊆X with property P such that X=St(A,U). We study the relationships of star P properties for P∈{Lindelöf,σ-compact, countable} with other Lindelöf type properties.

We study the subposet E3(X) of the lattice L1(X) of all T1-topologies on a set X, being the collections of all T3 topologies on X, with a view to deciding which elements of this partially ordered set have and which do not have immediate predecessors. We show that each regular topology which is not R-closed does have such a predecessor and as a corollary we obtain a result of Costantini that each non-compact Tychonoff space has an immediate predecessor in E3. We also consider the problem of when an R-closed topology is maximal R-closed.