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Let G be a locally compact first countable Abelian topological group of cardinality ≤ c and suppose that for every n ∈ N the subgroup nG is not totally bounded. We show that (1) the topology of G can be extended to a locally compact first countable semigroup topology T on S = G ⊕ ( ⊕ λ Z + ) for some λ ∈ [ p , c ] such that G is dense in T and ( S , T ) is absolutely closed in the class of cancellative topological semigroups with the Frechet-Urysohn property, and (2) assuming Martin's Axiom, the topology of G can be extended to a locally compact first countable semigroup topology T on S = G ⊕ ( ⊕ c Z + ) such that G is dense in T and ( S , T ) is absolutely closed in the class of all topological semigroups with the Frechet-Urysohn property.

The class of weakly linearly Lindelof spaces was introduced and studied by Juhasz, Tkachuk and Wilson in [7] . Recall that a space X is weakly linearly Lindelof if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x ∈ X such that every neighborhood of x meets κ-many elements of U . In this paper, we show that: (1) If X is a weakly linearly Lindelof space and U is an open cover of X, then for the open cover { St 2 ( x , U ) : x ∈ X } of X, there exists a countable subset A ⊂ X such that St 2 ( A , U ) ‾ = X ; (2) Every weakly linearly Lindelof normal metaLindelof space is weakly Lindelof; (3) If X is a first countable regular space, then M ( X ) (generated by Moore Machine) is weakly linearly Lindelof if and only if X is weakly linearly Lindelof; (4) Every product of a weakly linearly Lindelof space and a space of countable spread (or a separable space) is weakly linearly Lindelof; (5) If a subspace X ⊂ ω 1 ω is weakly linearly Lindelof, then X is second countable (and hence, metrizable); (6) If X is a weakly linearly Lindelof Baire space with a rank 2-diagonal such that w e ( X ) ≤ ω 1 , then | X | ≤ c ; (7) The space X is cellular-WLL if and only if it is weakly linearly Lindelof.

We show that if κ ≤ ω and there exists a group topology without non-trivial convergent sequences on an Abelian group H such that H n is countably compact for each n κ then there exists a topological group G such that G n is countably compact for each n κ and G κ is not countably compact. If in addition H is torsion, then the result above holds for κ = ω 1 . Combining with other results in the literature, we show that: a) Assuming c incomparable selective ultrafilters, for each n ∈ ω , there exists a group topology on the free Abelian group G such that G n is countably compact and G n + 1 is not countably compact. (It was already know for ω). b) If κ ∈ ω ∪ { ω } ∪ { ω 1 } , there exists in ZFC a topological group G such that G γ is countably compact for each cardinal γ κ and G κ is not countably compact.

Based on the notions of T. Banakh's strict Pytkeev networks and A.V. Arhangel'skii's sensor families, strict Pytkeev networks with sensors are introduced in this paper. A family P of subsets of a topological space X is called a strict Pytkeev network with sensors (abbr. an sp-network) if, for each x ∈ U ∩ A ‾ with U open and A subset in X, there is a set P ∈ P such that x ∈ P ⊂ U and x ∈ P ∩ A ‾ . In present paper, we discuss certain relationship and operations among spaces defined by special Pytkeev networks, study spaces with a point-countable sp-network and spaces with a σ-closure-preserving sp-network, and detect some applications of sp-networks in topological groups. The following results are obtained: (1) Every sp-network is preserved by a continuous pseudo-open mapping. (2) Every k-space with a point-countable sp-network coincides with a continuous pseudo-open s-image of a metric space. (3) Every regular feebly compact space with a point-countable sp-network has a point-countable base. (4) A regular space has a countable sp-network if and only if it is separable and has a point-countable sp-network. (5) A topological space is stratifiable if and only if it is a regular space with a σ-closure-preserving sp-network. (6) A regular space with a σ-locally finite sp-network has a σ-discrete sp-network. (7) A topological group is metrizable if it has countable sp-character. (8) There is a non-Frechet-Urysohn sequential topological group with a countable strict Pytkeev network, which give a negative answer to a question posed by A.V. Arhangel'skii [1] .

Star selection principles first introduced and studied by Kocinac et al. in [10] and [2] are investigated. In particular we consider the relationship between the star-Menger, strongly star-Menger as well as the relationship between star covering properties of the Niemytzki plane with the dominating number and the bounding number. Furthermore, we give a characterization of the strongly star-Menger property in terms of games on a strongly star-Lindelof space which consist of the disjoint union of a closed discrete set with a σ-compact subspace. A consistent example of a metacompact star-Menger not strongly star-Menger space is given. In addition, we show some results related to weaker properties than paracompactness and we study particular examples. We pose several problems about these properties.

We prove that in any Hausdorff space, the Rothberger game is equivalent to the k-Rothberger game, i.e. the game in which player II chooses k open sets in each move. This result follows from a more general theorem in which we show these games are equivalent to a game we call the restricted Menger game. In this game I knows immediately in advance of playing each open cover how many open sets II will choose from that open cover. This result illuminates the relationship between the Rothberger and Menger games in Hausdorff spaces. The equivalence of these games answers a question posed by Aurichi, Bella, and Dias [1] , at least in the context of Hausdorff spaces.

We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [6] holds while the classical Borel Conjecture fails.

Todorcevic introduced the forcing axiom PFA(S) and established many consequences. We contribute to this project. In particular, we consider status under PFA(S) of two important consequences of PFA concerning spaces of countable tightness. We prove that the existence of a Souslin tree does not imply the existence of a compact non-sequential space of countable tightness. We contrast this with M.E. Rudin's result that the existence of a Souslin tree does imply the existence of an S-space (and the later improvement by Dahrough to a compact S-space). On the other hand, PFA(S) implies there is a first-countable perfect pre-image of ω 1 that contains no copies of ω 1 .

We prove that every cellular-Lindelof space is weakly linearly Lindelof and hence monotonically normal cellular-Lindelof spaces must be Lindelof. We also show that there exists a cellular-Lindelof locally Lindelof P-space that is not weakly Lindelof. Under CH, we establish that every first countable normal weakly linearly Lindelof space is weakly Lindelof and has cardinality not exceeding c . Our results solve an open question published by Bella and Spadaro.

We investigate basic properties of three cardinal invariants involving ω 1 : stick, antichain number, and matching number. The antichain number is the least cardinal κ for which there does not exist a subcollection of size κ of uncountable subsets of ω 1 with pairwise finite intersections and the matching number is the least cardinal κ for which there exists a subcollection X of size κ of order-type ω subsets of ω 1 so that every uncountable subset of ω 1 has infinite intersection with a member of X. We demonstrate how these numbers are affected by Cohen forcing and also prove some results about the effect of Hechler forcing. We also introduce a forcing notion to increase the matching number, and study its basic properties.

We study the completeness and ultracompleteness numbers of a convergence space. In the case of a completely regular topological space, the completeness number is countable if and only if the space is $\v{C}$ech-complete, and the ultracompleteness number is countable if and only if the space is ultracomplete. We show that the completeness number of a space is equal to the pseudopaving number of the upper Kuratowski convergence on the space of its closed subsets, at $\emptyset$. Similarly, the ultracocompleteness number of a space is equal to the paving number of the upper Kuratowski convergence on the space of its closed subsets, at $\emptyset$., Comment: 17 pages

Given a topological property (or a class) P, the class P ′ consists of spaces X such that for any neighbourhood assignment ϕ on X, there exists a subspace Y ⊂ X with property P for which ϕ ( Y ) = ⋃ { ϕ ( y ) : y ∈ Y } is dense in X. The class P ′ are called the weak dual of P or weakly dually P (with respect to neighbourhood assignments). We establish that DCCC is weakly self-dual in the class of weakly regular spaces. If P ∈ { weakly Lindelof , CCC , separable } , then P is weakly self-dual in the class of Baire developable spaces. By using Erdos–Rado's theorem, we also prove that: (1) If X is a Baire, weakly dually CCC Hausdorff space with a rank 2-diagonal, then X has cardinality at most 2 ω ; (2) If X is a Baire, weakly dually DCCC Hausdorff space with a rank 3-diagonal, then X has cardinality at most 2 ω ; (3) If X is a weakly dually DCCC Hausdorff space with a rank 4-diagonal, then X has cardinality at most 2 ω .

We study, in a systematic manner, products of general topological spaces with Menger's covering property, and its refinements based on filters and semifilters. To this end, we apply Dedekind compactification to extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming the Continuum Hypothesis, every productively Lindelof space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.

Whenever P is a topological property, we say that a topological space X is star P if whenever U is an open cover of X, there is a subspace Y ⊂ X with property P such that St ( Y , U ) = X . We study the relationships of star P properties for P ∈ { DCCC , weakly Lindelof , CCC } with other chain conditions. By using Erdos–Rado's theorem, we also establish several cardinal inequalities for star CCC spaces.

We investigate the possibility of extension of fragmented functions from Lindelof subspaces of completely regular spaces and find necessary and sufficient conditions on a fragmented Baire-one function to be extendable on any completely regular superspace.

This manuscript extends both Cantor intersection theorem and Cantor–Kuratowski intersection theorem from the setting of metric spaces to the setting of admissible spaces. Completeness is revisited and the notion of uniformly locally compact admissible space is introduced. Compact, complete, and totally bounded admissible spaces are related.

Todorcevic introduced the forcing axiom PFA(S) and established many consequences. We contribute to this project. In particular, we show that forcing with the Souslin tree S, as postulated by PFA(S), will preserve that all automorphisms of the Boolean algebra P ( N ) / fin are trivial.

The class of cellular-Lindelof spaces was introduced and studied by A. Bella and S. Spadaro (2017) [4] . We say that a topological space X is cellular-Lindelof if for every family U of pairwise disjoint non-empty open sets of X there is a Lindelof subspace L ⊂ X such that U ∩ L ≠ ∅ , for every U ∈ U . In this paper, we first study topological properties of cellular-Lindelof spaces, and the relations between cellular-Lindelof spaces and related spaces. In particular, we obtain a Tychonoff example of a weakly Lindelof space which is not cellular-Lindelof, which gives a positive answer to a question of A. Bella and S. Spadaro ( [4, Question 2] ). We also prove that every monotonically normal W-space is cellular-Lindelof if and only if it is Lindelof. Finally, by using Erdos–Rado's theorem, we establish some cardinal inequalities for cellular-Lindelof spaces. Some new questions are also posed.

Motivated by results of Juhasz and van Mill in [13] , we define the cardinal invariant w t ( X ) , the weak tightness of a topological space X , and show that | X | ≤ 2 L ( X ) w t ( X ) ψ ( X ) for any Hausdorff space X (Theorem 2.8). As w t ( X ) ≤ t ( X ) for any space X , this generalizes the well-known cardinal inequality | X | ≤ 2 L ( X ) t ( X ) ψ ( X ) for Hausdorff spaces (Arhangel′skiĭ [1] , Sapirovskiĭ [17] ) in a new direction. Theorem 2.8 is generalized further using covers by G κ -sets, where κ is a cardinal, to show that if X is a power homogeneous compactum with a countable cover of dense, countably tight subspaces then | X | ≤ c , the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting.

We prove that a Hausdorff metaLindelof weakly star countable space is feebly Lindelof and a Hausdorff metacompact weakly star finite space is almost compact which partially answers a question of Alas and Wilson (2017) [2, Question 3.14] . We also obtain a normal example of a weakly star countable space which is neither almost star countable nor star Lindelof without any set-theoretic assumptions, which answers a question implicitly asked by Song (2015) [13, Remark 2.8] and a question asked by Alas, Junqueira and Wilson (2011) [3, Question 4] . Under MA+¬CH, there even exists a normal weakly star countable Moore space which is not almost star countable. An example of a Tychonoff star compact and weakly star finite space which is not star countable is also given. Finally, we prove that every weakly star countable Hausdorff space with a rank 4-diagonal has cardinality at most 2 ω .

It is proved in ZF that if a collection of (quasi)-metric spaces is indexed by a countable union of finite sets, then the product of this collection is (quasi)-metrizable, while it is independent of ZF that every countable product of metrizable spaces is quasi-metrizable. It is also proved that if J is a non-empty set, while a space X, consisting of at least two points, is equipped with the co-finite topology, then the product X J is quasi-metrizable if and only if both X and J are countable unions of finite sets. Several equivalences of CUT ( fin ) are deduced. It is shown that, in every model of ZF + ¬ CUT ( fin ) , for an uncountable set J, the space N J can be normal (even metrizable), a Cantor cube can be simultaneously metrizable, not second-countable and non-compact.

Balogh proved that PFA , the Proper Forcing Axiom, implies that a countably compact space with countable tightness is either compact or contains an uncountable free sequence. Eisworth established the relevance to proper forcing of strengthening the countable tightness assumption to that of hereditary countable π-character. Eisworth proved that for any countably compact space with hereditary countable π-character there is a totally proper forcing that adds an uncountable free sequence. We extend these results by showing that PFA implies that countably compact spaces are closed in spaces that have hereditary countable π-character. This gives a countably compact version of the Moore–Mrowka problem in the class of spaces with hereditary countable π-character.

In this paper, we prove that DCCC is self-dual with respect to neighborhood assignments. Some related results on dually CCC (or dually DCCC) spaces are also obtained. Moreover, we prove that the cardinality of a dually CCC space X is at most 2 ω if X satisfies one of the following conditions: (1) X has a rank 2-diagonal; (2) X is first countable and has a G δ -diagonal; (3) X is first countable and perfect. Finally, we prove that the extent of a first countable dually CCC space is at most 2 ω .

In [4] , it is asked whether orthocompact products of two GO-spaces are normal or not. In this paper, we see that a product of a GO-space and a countably compact GO-space is normal iff it is orthocompact. Also we prove that a normal product of a GO-space and a countably compact GO-space is countably paracompact.

In this paper, we shall consider the hyperspace of all nontrivial convergent sequences S c ( X ) of a Frechet-Urysohn nondiscrete space X, which is equipped with the Vietoris topology. We study the spaces X for which S c ( X ) is Baire: this kind of spaces have a dense subset of isolated points (see [12] ). We characterize the spaces X for which S c ( X ) is Baire. This characterization uses a topological game inspired by the Banach-Mazur game. As a consequence, we obtain that if X is completely metrizable and has a dense subset of isolated points, then S c ( X ) is Baire. Our last main result shows that S c ( X ) is pseudocompact iff X is homeomorphic to the one-point compactification of a discrete space of uncountable size. This last assertion provides a characterization of the one-point compactification of a discrete space of uncountable size.

Tukey order is used to compare the cofinal complexity of partially order sets (posets). We prove that there is a 2 c -sized collection of sub-posets in 2 ω which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of ω ω is a Tukey quotient of ω ω , we answer two open questions published in [14] . The relation between P-base and strong Pytkeev⁎ property is investigated. Let P be a poset equipped with a second-countable topology in which every convergent sequence is bounded. Then we prove that any topological space with a P-base has the strong Pytkeev⁎ property. Furthermore, we prove that every uncountably-dimensional locally convex space (lcs) with a P-base contains an infinite-dimensional metrizable compact subspace. Examples in function spaces are given.

We force a classification of all the Abelian groups of cardinality at most 2 c that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov [10] for cardinality at most 2 c , by showing that if a non-torsion Abelian group of size at most 2 c admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.

Assuming the existence of c incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality c that admit a countably compact group topology. We show that for each κ ∈ [ c , 2 c ] each of these groups has a countably compact group topology of weight κ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of 2 c selective ultrafilters, there are at least 2 c non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most 2 c is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality c that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.

We try to find a common extension of two cardinal inequalities for Lindelof spaces. Using an estimate of the number of G δ points due to Balogh, we improve a result of Juhasz and Spadaro. A cardinal inequality for linearly Lindelof 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.

We introduce a σ -ideal on ω 1 × ω 1 and a filter on the collection of graphs of strictly decreasing partial functions on ω 1 taking values in ω 1 . We use them to prove that a certain space is a non-bisequential Eberlein compactum.

We show that, for a Lindelof Σ-space X, if C p ( X , [ 0 , 1 ] ) is strongly dominated by a second countable space, then X is countable. Under Martin's Axiom we prove that there exists a countable space Z that strongly dominates the complement of the diagonal of any first countable compact space. In particular, strong domination by a countable space of the complement of the diagonal of a compact space X need not imply metrizability of X. It turns out that the same countable space Z strongly dominates C p ( X ) for an uncountable space X. Our results solve several published open problems.

The study of the class of van der Waerden rings was initiated by Comfort, Remus and Szambien in [7] . A compact ring ( R , T ) is a van der Waerden ring if and only if T is the only totally bounded ring topology on R . In this paper the study of this class of compact rings is continued. We prove that an Abelian compact van der Waerden ring R has the form R = ( ∏ i ∈ I R i ) × L , where L is a compact radical ring with radical topology and R i ( i ∈ I ) is a compact local topologically finitely generated ring. It is shown that every van der Waerden ring is totally disconnected and metrizable. Furthermore, the class of van der Waerden rings is closed under extensions. Different classes of compact rings are studied which are closely related to van der Waerden rings.

A classical theorem of Malykhin says that if { X α : α ≤ κ } is a family of compact spaces such that t ( X α ) ≤ κ , for every α ≤ κ , then t ( ∏ α ≤ κ X α ) ≤ κ , where t ( X ) is the tightness of a space X . In this paper we prove the following counterpart of Malykhin's theorem for functional tightness: Let { X α : α λ } be a family of compact spaces such that t 0 ( X α ) ≤ κ for every α λ . If λ ≤ 2 κ or λ is less than the first measurable cardinal, then t 0 ( ∏ α λ X α ) ≤ κ , where t 0 ( X ) is the functional tightness of a space X . In particular, if there are no measurable cardinals, then the functional tightness is preserved by arbitrarily large products of compacta. Our result answers a question posed by Okunev.

A topological space $X$ has the strong Pytkeev property at a point $x\in X$ if there exists a countable family $\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\subset X$ and subset $A\subset X$ accumulating at $x$, there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. We prove that for any $\aleph_0$-space $X$ and any space $Y$ with the strong Pytkeev property at a point $y\in Y$ the function space $C_k(X,Y)$ has the strong Pytkeev property at the constant function $X\to \{y\}\subset Y$. If the space $Y$ is rectifiable, then the function space $C_k(X,Y)$ is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces $(X_n,*_n)$, $n\in\omega$, with the strong Pytkeev property their Tychonoff product and their small box-product both have the strong Pytkeev property at the distinguished point. We prove that a sequential rectifiable space $X$ has the strong Pytkeev property if and only if $X$ is metrizable or contains a clopen submetrizable $k_\omega$-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1311.1468

We show that for a number of cardinal functions ϕ, an ω-narrow topological group G admits a continuous homomorphism onto a topological group H satisfying ϕ(H)=w(H)=τ, for every infinite regular cardinal τ≤ϕ(G). In particular, the conclusion is valid when ϕ is weight, cellularity, character, pseudocharacter, tightness or o-tightness (the regularity of τ is not necessary for the cellularity). We also show that both pseudocompactness and precompactness reflect in continuous homomorphic images of countable weight, so an ω-narrow topological group is pseudocompact (precompact) if and only if all continuous second countable homomorphic images of the group are pseudocompact (precompact). It is established that the Baire property of ω-narrow groups reflects in continuous homomorphic images of weight less than or equal to the continuum. It turns out, however, that neither compactness, countable compactness, nor (weak) Lindelofness reflects in continuous homomorphic images of any ‘small’ weight.

Hurewicz proved completely metrizable Menger spaces are σ-compact. We extend this to Cech-complete Menger spaces, and consistently, to projective metrizable Menger spaces. On the other hand, it is consistent that there is a co-analytic Menger metrizable space that is not σ-compact.

A space X is selectively absolutely star-Lindelof if for any open cover U of X and any sequence ( D n : n ∈ ω ) of dense subsets of X, there are finite sets F n ⊂ D n ( n ∈ ω ) such that S t ( ⋃ n ∈ ω F n , U ) = X . This notion was introduced by S. Bhowmik [3] , and it lies between absolute countable compactness in Matveev [9] and absolute star-Lindelofness in Bonanzinga [4] . In this paper, we distinguish absolute star-Lindelofness from selective absolute star-Lindelofness, and study the general properties of selectively absolutely star-Lindelof spaces.

We show that neither first-countable nor second-countable are three-space properties in the class of paratopological groups: we present a countable regular paratopological Abelian group H which contains a closed discrete subgroup F such that H / F is topologically isomorphic to the rational numbers with the Sorgenfrey topology and H is not first-countable. Also, we prove that if H is an invariant topological subgroup of a paratopological group G such that H is second-countable and G / H has countable network, then G has countable network as well (this answers a question posed in [12] ). Hence if H is an invariant topological subgroup of a first-countable paratopological group G such that H and G / H are second-countable, then so is G.

The main purpose of this note is to prove that the product of a cellular-Lindelof space with a space of countable spread need not be cellular-Lindelof.

In this paper we extend the notion of a clopen subset of a topological space to that of a clopen subobject in a topological category. We relate clopen subobjects to the concept of normalized topological category via a theorem which, in turn, shows why certain subsets must be clopen in a topological space. We employ clopen subobjects to extend the topological notion of connectedness to topological categories. Examples of clopen subobjects and connected objects in several topological categories are given.

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] .

We present a series of axioms that describe the combinatorial core of the Mathias model (countable support iteration of ω 2 Mathias reals over the Continuum Hypothesis). Our axioms are formulated in terms of games with Borel sets and functions and are strong enough to imply most of the propositions usually proved by means of the iterated Mathias forcing.

Given a hereditarily meager ideal I on a countable set X we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology τ I on X such that τ I ∩ I = { ∅ } and, moreover, if I is p + then τ I is selectively separable (SS) and if I is q + , so is τ I . In particular, we obtain regular maximal spaces satisfying all boolean combinations of the properties SS and q + .

We establish that, for any Tychonoff space X, at least one of the function spaces C p C p ( X ) and C p C p C p ( X ) must have a dense subspace of countable pseudocharacter. If GCH holds, then at least one of the spaces C p ( X ) and C p C p ( X ) has a dense subspace of countable pseudocharacter. We also prove that all iterated function spaces C p , n ( K ) have a uniformly dense subspace of countable pseudocharacter whenever K is a Corson compact space. Our results solve several published open questions.

We prove that for every fixed nondegenerate Peano continuum X there exists a continuum-sized family of compactifications of the ray with X as remainder that is pairwise incomparable by continuous mappings.

In this paper, we prove that the cardinality of a star Lindelof space X does not exceed c if X satisfies one of the following conditions: (1) X has a rank 3-diagonal; (2) X is normal and has a rank 2-diagonal; (3) X is first countable, normal and has a G δ -diagonal. Moreover, we also obtain several results concerning the general question “When must a star Lindelof space be star countable?”.

By introducing the class of coideals on an infinite directed set X, a class that contains all the adequate ultrafilters on X, it becomes possible to prove partition results for the ordered finite or infinite sequences in X with respect to a given coideal on X. Our theory extends the classical topological Ramsey theory, and in addition includes as particular cases (a) the corresponding theory for coideals on the set of natural numbers proved by Louveau, Mathias, Farah and Todorcevic, (b) the Milliken–Taylor partition theorems for sequences of finite subsets of natural numbers, and (c) the partition theorems for sequences of words proved by Carlson, Bergelson–Blass–Hindman, Farmaki.

We consider combinatorial versions of classical separation axioms, give various examples of spaces satisfying them, consider interrelations between them and also how classical results including separation axioms extend to these new notions. We also consider their role in restricting cardinality of topological spaces and pose several open problems.

In this note we show that a T 1 topological space X is a Menger space if and only if for each sequence { ϕ n : n ∈ ω } of neighborhood assignments for X, there exists, for each n ∈ ω , a finite subset D n of X such that X = ⋃ { ϕ n ( d ) : d ∈ D n , n ∈ ω } and D = ⋃ { D n : n ∈ ω } is a closed discrete subspace of X. A T 1 topological space X is a Rothberger space if and only if for each sequence { ϕ n : n ∈ ω } of neighborhood assignments for X, there exists, for each n ∈ ω , a point d n ∈ X such that X = ⋃ { ϕ n ( d n ) : n ∈ ω } and D = { d n : n ∈ ω } is a closed discrete subspace of X. Let M be the class of all spaces with the Menger property. Let R be the class of all Rothberger spaces. We show that a M-like space has the Menger property. A R-like space is a Rothberger space. Let C be the class of all compact spaces. Let W be the class of all countable spaces. We prove that a finite product of C-like spaces is a C-like space. As a corollary we know that a finite product of C-like spaces is a Menger space. In the last part of this note we show that a finite product of W-like Hausdorff spaces is a nc-W-like space. A finite product of W-like spaces is a Rothberger space.

We prove that it is consistent (even with Martin's Axiom) that there is first-countable initially ω 1 -compact space with cardinality greater than the continuum. We also prove that it is consistent with Martin's Axiom and c = ω 2 that there is a compact space of countable tightness which is not sequential. It is known that neither statement is consistent with the Proper Forcing Axiom. We use an innovative new method of constructing proper posets with elementary submodels as side conditions introduced by Neeman.