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Let I be an ideal on ω. Following Baumgartner (1995) [2] , we say that an ultrafilter U on ω is an I -ultrafilter if for every function f : ω → ω there is A ∈ U with f [ A ] ∈ I. In particular, P-points are exactly Fin × Fin -ultrafilters. If there is an I -ultrafilter which is not a J -ultrafilter, then I is not below J in the Katětov order ⩽ K (i.e., for every function f : ω → ω there is A ∈ I with f − 1 [ A ] ∉ J), however the reversed implication is not true (even consistently). Recently it was shown that for all Borel ideals I we have: I ≰ K Fin × Fin if and only if in some forcing extension one can find an I -ultrafilter which is not a P-point (Filipów et al. (2022) [6]). We show that under some combinatorial assumptions imposed on the ideal J , the classes of J -ultrafilters and Fin × J -ultrafilters coincide. This allows us to find some sufficient conditions on ideals to obtain the equivalence: I ≰ K Fin × J if and only if in some forcing extension one can find an I -ultrafilter which is not a J -ultrafilter. We provide several examples of ideals, for which the above equivalence is true, including the ideal of nowhere dense subsets of Q and the ideal of sets of asymptotic density zero. [ABSTRACT FROM AUTHOR]

For a T 1 -space X , we denote by P R (X) the Pixley-Roy space over X. For p ∈ ω ⁎ , let X p = { p } ∪ ω be the subspace of the Stone-Čech compactification βω of the discrete space ω. Motivated by Gul'ko's theorem (Theorem 1.1), we show: (1) P R (X p) and P R (X q) are homeomorphic if and only if X p and X q are homeomorphic (i.e., p and q are type-equivalent), (2) if q is selective and X q can be embedded into P R (X p) , then X p and X q are homeomorphic, (3) if p is selective, then P R (X p) contains copies of some X q n (n ∈ N) which are pairwise non-homeomorphic, and (4) P R (X p) , P R (X p ⊕ X p) and P R (X p ⁎ X p) are pairwise non-homeomorphic, where X p ⁎ X p is the quotient space obtained by identifying the limit points of the topological sum X p ⊕ X p. [ABSTRACT FROM AUTHOR]

This article studies different topological properties of the space of maximal d -elements of an M -frame with a unit. We characterize when the space M a x (d L) is Hausdorff, answering the question posed in [2]. We also characterize other topological properties of M a x (d L) , namely zero-dimensional, discrete, and clopen π -base. The concept of weak-component elements is introduced here, as a generalized idea from the theory of rings, which is essential in the study of d -semiprime frames. [ABSTRACT FROM AUTHOR]

A Michael space is a Lindelöf space which has a non-Lindelöf product with the Baire space. In this work, we present the notion of Michael ultrafilter and we use it to construct a Michael space under the existence of a selective ultrafilter and max ⁡ { b , g } = d. [ABSTRACT FROM AUTHOR]

We work in set theory without the Axiom of Choice (AC) and establish the following results: 1. "Products of ultracompact spaces are ultracompact" + "there exists a free ultrafilter on ω " (UF (ω)) is equivalent to AC in ZFA ; 2. "Products of ultracompact spaces are ultracompact" does not imply AC in ZF ; 3. AC is equivalent to each of "products of ultracompact spaces are countably compact", "products of ultracompact spaces are sequentially limit point compact", "products of ℵ 0 -bounded spaces are countably compact", "products of ℵ 0 -bounded spaces are sequentially limit point compact", and "products of ℵ 0 -bounded spaces are ℵ 0 -bounded" in ZFA ; 4. UF (ω) is equivalent to each of "every ultracompact space is sequentially limit point compact" and "every ultracompact T 2 space is sequentially limit point compact"; 5. The Axiom of Countable Choice (AC ω) is equivalent to "every ℵ 0 -bounded space is countably compact"; 6. AC ω + UF (ω) is equivalent to each of "every ultracompact space is countably compact", "products of countably many ultracompact spaces are countably compact", "products of countably many ultracompact spaces are sequentially limit point compact", and "products of countably many ℵ 0 -bounded spaces are ℵ 0 -bounded"; 7. Each of "a T 3 space is ultracompact, if and only if, it is ℵ 0 -bounded" and "a Tychonoff space is ultracompact, if and only if, it is ℵ 0 -bounded" is equivalent to "every filter on ω can be extended to an ultrafilter on ω " (BPI (ω)), and thus each of the above statements is equivalent to the statement "the Stone space βω of all ultrafilters on ω is compact". Thus we provide the exact characterizations of the above topological results, the former by Vaughan in "Countably compact and sequentially compact spaces", and the latter by Bernstein in "A new kind of compactness for topological spaces", as a specific renowned weak choice principle (which is strictly weaker than the Boolean Prime Ideal Theorem in ZF); 8. AC ω implies Ginsburg's theorem in "Some results on the countable compactness and pseudocompactness of hyperspaces", namely "a T 2 space X is ultracompact, if and only if, the hyperspace C L (X) (the set of non-empty closed subsets of X with the Vietoris topology) is ultracompact", which in conjunction with UF (ω) implies the weak choice principle A C fin ω (AC restricted to countable families of non-empty finite sets). [ABSTRACT FROM AUTHOR]

Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category LKHaus of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category CLCA of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence LKHaus ≃ CLCA op that was obtained by the first author more than a decade ago. Unlike the morphisms of CLCA , the morphisms of the new category and their composition law are very natural and easy to handle. [ABSTRACT FROM AUTHOR]

For a discrete group G , we use the natural correspondence between ideals in the Boolean algebra P G of subsets of G and closed subsets in the Stone–Čech compactification βG as a right topological semigroup to introduce and characterize some new ideals in βG . We show that if a group G is either countable or Abelian then βG contains no ideals that are maximal among the closed proper subideals of G ⁎ , G ⁎ = β G ∖ G , but this statement does not hold for the group S κ of all permutations of an infinite cardinal κ . We characterize the minimal closed ideal in βG containing all idempotents of G ⁎ . [ABSTRACT FROM AUTHOR]

Lindelöf numbers of sets Z ⊂ β (λ) ∖ λ are examined in G δ -topologies (and in G μ -topologies). The results are in a close connection to measurable and μ -strongly compact cardinal numbers. Those numbers are characterized by means of extensions of weakly complete filters, by τ -compactness of sets of certain complete ultrafilters and by Lindelöf numbers of the set of uniform ultrafilters in G μ -topologies. Some known results about μ -strongly compact cardinals are reproved using set-theoretical methods. [ABSTRACT FROM AUTHOR]

For any composant E ⊂ H ⁎ and corresponding near-coherence-class E ⊂ ω ⁎ we prove the following are equivalent: (1) E properly contains a dense semicontinuum. (2) Each countable subset of E is contained in a dense proper semicontinuum of E . (3) Each countable subset of E is disjoint from some dense proper semicontinuum of E . (4) E has a minimal element in the finite-to-one weakly-increasing order of ultrafilters. (5) E has a Q -point. A consequence is that NCF is equivalent to H ⁎ containing no proper dense semicontinuum and no non-block points. This gives an axiom-contingent answer to a question of the author. Thus every known continuum has either a proper dense semicontinuum at every point or at no points. We examine the structure of indecomposable continua for which this fails, and deduce they contain a maximum semicontinuum with dense interior. [ABSTRACT FROM AUTHOR]

We study the partition properties enjoyed by the “next best thing to a P-point” ultrafilters introduced recently in joint work with Dobrinen and Raghavan. That work established some finite-exponent partition relations, and we now analyze the connections between these relations for different exponents and the notion of conservativity introduced much earlier by Phillips. In addition, we establish some infinite-exponent partition relations for these ultrafilters and also for sums of non-isomorphic selective ultrafilters indexed by selective ultrafilters. [ABSTRACT FROM AUTHOR]

This article introduces a line of investigation into connections between creature forcings and topological Ramsey spaces. Three examples of sets of pure candidates for creature forcings are shown to contain dense subsets which are actually topological Ramsey spaces. A new variant of the product tree Ramsey theorem is proved in order to obtain the pigeonhole principles for two of these examples. [ABSTRACT FROM AUTHOR]

We investigate the question of whether Q carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this question in three different ways: by assuming a certain parametrized diamond principle, from the cardinal invariant equality d = c , and in the Random real model. [ABSTRACT FROM AUTHOR]

We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting. Furthermore, we discuss the relation with Tukey reducibility and prove that, assuming the Continuum Hypothesis, there exist ultrafilters on the Cohen algebra which are RK-equivalent in the generalized sense but Tukey-incomparable, in stark contrast with the classical setting. [ABSTRACT FROM AUTHOR]

This is a survey describing Mary Ellen Rudin's theorem about the number of composants of the Stone–Čech remainder of a half-line and subsequent related work, especially the principle of near coherence of filters. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a countably infinite discrete group and let βG be the Stone–Čech compactification of G. Let denote the finest decomposition of into closed left ideals of βG with the property that the corresponding quotient space of is Hausdorff, and the finest decomposition of into closed left ideals of βG. We show that there is a dense subset of points such that . In particular, is finer than . [Copyright &y& Elsevier]

Abstract: Let X be an unbounded metric space, for all and . We endow X with the discrete topology and identify the Stone–Čech compactification βX of X with the set of all ultrafilters on X. Our aim is to reveal some features of algebra in βX similar to the algebra in the Stone–Čech compactification of a discrete semigroup [6]. We denote and, for , write if and only if there is such that for each , where . A subset is called invariant if and imply . We characterize the minimal closed invariant subsets of X, the closure of the set , and find the number of all minimal closed invariant subsets of . For a subset and , we denote and say that a subset is an ultracompanion of Y if for some . We characterize large, thick, prethick, small, thin and asymptotically scattered spaces in terms of their ultracompanions. [Copyright &y& Elsevier]

Abstract: Let G be a countably infinite group. A topology on G is left invariant if left translations are continuous. A left invariant topology is strongly complete if it is regular and for every partition of G into open sets, there is a neighborhood V of 1 such that for every , is finite. We show that assuming MA, for every , there is a strongly complete left invariant topology on G with exactly n nonprincipal ultrafilters converging to 1, and in the case , can be chosen to be a group topology. We also show that it is consistent with ZFC that if G can be embedded algebraically into a compact group, then there are no such topologies on G. [Copyright &y& Elsevier]

Abstract: Let G be a countably infinite discrete group, let βG be the Stone–Čech compactification of G, and let . Let denote the finest decomposition of into closed left ideals of βG with the property that the corresponding quotient space of is Hausdorff. We show that it is consistent with ZFC that, if G can be embedded algebraically into a compact group, then every contains maximal principal left ideals of βG. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss properties of topological spaces with algebraic structures and answer several problems posed in [A.V. Arhangelʼskii, M. Tkachenko, Topological Groups and Related Structures, Atlantics Press/World Sci., 2008]. We consider a topology on an infinite discrete group G generated by a free ultrafilter p on G and show that this topology can be Hausdorff in the case when G is the group of integers, even if p is not an idempotent. Two open continuous homomorphisms f of a paratopological group G onto a paratopological group H are constructed such that: (a) H is paracompact and the kernel of f is locally compact, but f is not locally perfect and G is not locally paracompact; (b) H and the kernel of f are metrizable, but G is not metrizable. We also show that every first-countable ω-narrow semitopological group is separable. [Copyright &y& Elsevier]

Abstract: Let be a Hausdorff semitopological semigroup, S be a dense subsemigroup of T and e be an idempotent element of T. The set of ultrafilters on S that converge to e is a semigroup under restriction of the usual operation + on , the Stone–Čech compactification of the discrete semigroup . We characterize the smallest ideal of , and those sets “central” in , that is, those sets which are members of minimal idempotents in . We describe some combinatorial applications of those sets that are central in . [Copyright &y& Elsevier]

Abstract: We establish negative results about “rectangular” local bases in compacta. For example, there is no compactum where all points have local bases of cofinal type . For another, the compactum βω has no nontrivially rectangular local bases, and the same is consistently true of : no local base in βω has cofinal type if for some . Also, CH implies that every local base in has the same cofinal type as one in βω. We also answer a question of Dobrinen and Todorčević about cofinal types of ultrafilters: the Fubini square of a filter on ω always has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on ω is commutative modulo cofinal equivalence. [Copyright &y& Elsevier]

Abstract: Given a dynamical system with X a compact metric space and a free ultrafilter p on , we define for all . It was proved by A. Blass (1993) that is recurrent iff there is such that . This suggests to consider those points for which for some , which are called p-recurrent. We shall give an example of a recurrent point which is not p-recurrent for several . Also, A. Blass proved that two points are proximal iff there is such that (in this case, we say that x and y are p-proximal). We study the properties of the p-proximal points of the following continuous self maps of the Cantor set: For an arbitrary function , we define by for every and for every (the shift map on is obtained by the function ). Let denote the Ellis semigroup of the dynamical system . We prove that if is a function with at least one infinite orbit, then is homeomorphic to . Two functions are defined so that is homeomorphic to the Cantor set, and is the one-point compactification of with the discrete topology. [Copyright &y& Elsevier]

Abstract: Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martinʼs Axiom for countable posets, to distinguish non-principal ultrafilters on ω up to homeomorphism. Here, we identify ultrafilters with subpaces of in the obvious way. Using the same methods, still under Martinʼs Axiom for countable posets, we will construct a non-principal ultrafilter such that is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial results about the relation of such topological properties with the combinatorial property of being a P-point. [Copyright &y& Elsevier]

Abstract: This paper contributes to the algebraization of topology via the theory of monads and lax extensions of monads and their associated lax algebras (see Barr (1970) , Clementino and Hofmann (2003) , Clementino, Hofmann and Tholen (2004) , Clementino and Tholen (2003) , Lowen and Vroegrijk (2008) , Manes (1974) , Seal (2005) ). We construct a monad , a lax extension and monad morphisms into from the most important monads as studied in the aforementioned papers such that their lax extensions and their associated categories of lax algebras can be derived from the extension by initial lifts via these monad morphisms. This provides us with a completely unified way to obtain the categories , , and without the necessity to leave the realm of as was previously required in Clementino and Hofmann (2003) , Clementino, Hofmann and Tholen (2004) and Clementino and Tholen (2003) in particular in order to obtain and . [Copyright &y& Elsevier]

Abstract: Let κ be an infinite cardinal. A uniform ultrafilter p on κ is a -point if for every decreasing κ-sequence of members of p, there is such that for all . We show that the existence of a topological group with certain extremal properties implies the existence of a -point for some κ. [Copyright &y& Elsevier]

Abstract: Let S be an infinite discrete semigroup which can be embedded algebraically into a compact topological group and let βS be the Stone–Čech compactification of S. We show that the smallest ideal of βS is not closed. [Copyright &y& Elsevier]

Abstract: We consider Ramsey-style partition theorems in which homogeneity is asserted not for subsets of a single infinite homogeneous set but for subsets whose elements are chosen, in a specified pattern, from several sets in prescribed ultrafilters. We completely characterize the sequences of ultrafilters satisfying such partition theorems. (Non-isomorphic selective ultrafilters always work, but, depending on the specified pattern, weaker hypotheses on the ultrafilters may suffice.) We also obtain similar results for analytic partitions of the infinite sets of natural numbers. Finally, we show that the two P-points obtained by applying the maximum and minimum functions to a union ultrafilter are never nearly coherent. [Copyright &y& Elsevier]

Abstract: For amenable groups there are “correspondence principles” relating the behavior under group translation for sets of positive density to that of sets of positive invariant measure for ergodic actions of the group. When the group is also “minimally almost periodic” such actions are automatically weakly mixing; hence the name WM groups. We study here the combinatorial consequences of this state of affairs. [Copyright &y& Elsevier]

Abstract: In this article we establish some connections between properties of a topological space X and the structure of convergence at -points in an arbitrary remainder of this space X. A special attention is given to the case when X is a topological group. [Copyright &y& Elsevier]

Abstract: Two non-discrete Hausdorff group topologies , on a group G are called transversal if the least upper bound of and is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits pairwise transversal complete group topologies on G (so maximal group topologies). Moreover, every abelian group G admits pairwise transversal complete group topologies. [Copyright &y& Elsevier]

Abstract: As applications of productivity of coreflective classes of topological spaces, the following results will be proved: (1) Characters of points of are not smaller than any submeasurable cardinal less or equal to . (2) If κ is a submeasurable cardinal and S is a sequential fan with κ many spines then the tightness of the κ-power of S is equal to κ. In fact, a little more general results are proved. [Copyright &y& Elsevier]

Abstract: Let denote the lattice consisting of the set of all quasi-uniformities on a set X, ordered by set-theoretic inclusion ⊆. We observe that a quasi-uniformity on X is the supremum of atoms of if and only if it is totally bounded and transitive. Each quasi-uniformity on X that is totally bounded or has a linearly ordered base is shown to be the infimum of anti-atoms of . Furthermore, each quasi-uniformity on X such that the topology of the associated supremum uniformity is resolvable has the latter property. [Copyright &y& Elsevier]

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]

This is a survey of four problems that are "classics" in many different senses of the word, and of several related problems associated with each one. The numbering of the Classic Problems picks up where that of a similar article left off about four decades ago: IX. Is every point of ω ⁎ a butterfly point? X. Is there a nonmetrizable perfectly normal, locally connected continuum? XI. Is there a normal space with a σ-disjoint base that is not paracompact? XII. Is there a regular symmetrizable space with a non- G δ point? Several related problems are given for each classic problem. Consistency results are summarized, and there is a discussion of each problem that explains various implications among the related problems and justifies calling certain problems equivalent. For each classic problem, an appendix goes deeper into some implications and/or includes reminisces. There is a purely set-theoretic problem related to Classic Problem IX. Call a filter on a set D nowhere maximal if it does not trace an ultrafilter on any subset of D. Related Problem D. Is every free ultrafilter the join of two nowhere maximal filters? It is shown that the special case of ultrafilters on ω is actually equivalent to Classic Problem IX. [ABSTRACT FROM AUTHOR]