The primary objective of this work is to construct spaces that are "pseudocompact but not countably compact", abbreviated as P-NC, while endowing them with additional properties. First, motivated by an old problem of van Douwen concerning first countable P-NC spaces with countable extent, we construct from CH a locally compact and locally countable first countable P-NC space with countable spread. A space is deemed densely countably compact, denoted as DCC for brevity, if it possesses a dense, countably compact subspace. Moreover, a space qualifies as densely relatively countably compact, abbreviated as DRC, if it contains a dense subset D such that every infinite subset of D has an accumulation point in X. A countably compact space is DCC, a DCC space is DRC, and a DRC space is evidently pseudocompact. The Tychonoff plank is a DCC space but is not countably compact. A 'If-space belongs to the class of DRC spaces but is-DCC. Lastly, if p E omega* is not a P-point, then T(p), representing the type of pin omega*, constitutes a pseudocompact subspace of omega* that is-DRC. When considering a topological property denoted as Q, we define a space X as "R-hereditarily Q" if every regular closed subspace of X also possesses property Q. The Tychonoff plank and the 'If-space are not R-hereditary examples for separating the above-mentioned properties. However, the aforementioned space T(p) is an Rhereditary example, albeit not being first countable. In this paper we want to find (first countable) examples which separate these properties R-hereditarily. We have obtained the following result. (1) There is a R-hereditarily "DCC, but not countably compact" space. (2) If CH holds, then there is a R-hereditarily "DRC, but-DCC" space. (3) If s = c, then there is a first countable, R-hereditarily "pseudo compact, but-DRC" space. In contrast to (2), it is unknown whether a first countable, R-hereditarily "DRC, but-DCC" space X can exist. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A base of a topological space is called Noetherian iff it does not contain an infinite strictly ⊆-increasing chain. We show that minimal cardinality of a regular spaces without a Noetherian base is the first strongly inaccessible cardinal, answering a question from the 1980s. We also study the Noetherian type of a topological space X, denoted by Nt(X), defined as the least cardinal κ such that X has a base ℬ with |{B'∈ℬ: B⊂ B'}|<κ for each B∈ℬ. The behavior of the Noetherian type under the G_δ-modification was investigated by Milovich and Spadaro. A central question, posed by them, is whether the Noetherian type of the G_δ-modification of the space D(2)^ℵ_ω is ω_1. This statement, denoted (Nt), is known to be independent of ZFC + GCH: it holds under “GCH + □_ℵ_ω”, but fails under “GCH + (ℵ_ω+1, ℵ_ω)→ (ℵ_1, ℵ_0)”. We place this phenomenon in a broader context by identifying similar independence phenomena for several topological and combinatorial principles. These include: (wFN) the weak Freese-Nation property of [ℵ_ω]^ω; (SAT) the existence of a saturated MAD family in [ℵ_ω]^ω; (HnT) the existence of an ω-homogeneous, but not ω-transitive permutation group on ℵ_ω; and (SPL) the existence of a countably compact, locally countable, and ω-fair regular space of cardinality ℵ_ω+1. Assuming GCH, we analyze the logical relationships between these principles and show, for example, that SPL implies wFN, which in turn implies both SAT, HnT and Nt, while SAT does not imply Nt.

One of the main results of the paper mentioned in the title says that from having 0

Hart and Kunen in [5] and, independently, R & iacute;os-Herrej & oacute;n in [14] defined and studied the class C(omega 1) of topological spaces X having the property that for every neighborhood assignment {U(y) : y E Y } with Y E [X]omega 1 there is Z E [Y]omega 1 such that Z subset of {U(z) : z E Z}. It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these two and several other natural classes that are sandwiched between them, thus clarifying some of the main problems that were raised in [14]. In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in C(omega 1) has countable net weight, answering a question that was raised by Hart and Kunen in both [5] and [6]. (c) 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org /licenses /by-nc-nd /4 .0/).

The notion of "pseudocompactness" was introduced by Hewitt. The concept of relatively countably compact subspaces were explored by Marjanovic to show that a Ψ-space is pseudocompact. A topological space is said to be DRC (DRS) iff it possesses a dense, relatively countably compact (or relatively sequentially compact, respectively) subspace. The concept of selectively pseudocompact game Sp(X) and the selectively sequentially pseudocompact game Ssp(X) were introduced by Dorantes-Aldama and Shakhmatov. They explored the relationship between the existence of a winning strategy and a stationary winning strategy for player P in these games. In particular, they observed that there exists a stationary winning strategy in the game Sp(X) (Ssp(X)) for Player P iff X is DRC (or DRS, respectively). In this paper we introduce natural weakening of the properties DRC and DRS: a space X is DRCo ( DRSo) iff there is a sequence (D_n:n ∈ω) of dense subsets of X such that every sequence (d_n:n ∈ω ) with d_n ∈ D_n has an accumulation point (or contains a convergent subsequence, respectively). These properties are also equivalent to the existence of some limited knowledge winning strategy on the corresponding games Sp(X) and Ssp(X). Clearly, DRS implies DRC and DRSo, DRC or DRSo imply DRCo. The main part of this paper is devoted to prove that apart from these trivial implications, consistently there are no other implications between these properties.

A novel selection principle was introduced by Dorantes-Aldama and Shakhmatov: a topological space X is termed selectively pseudocompact if for any sequence (U_n:n∈ ω) of pairwise disjoint non-empty open sets of X, one can choose points x_n∈ U_n such that the sequence (x_n:n∈ ω) has an accumulation point. In this paper, we explore various versions of this principle when we permit the selection of finite, scattered, or nowhere dense sets instead of just singletons. We develop a method to prove that the aforementioned versions of selective pseudocompactness are indeed distinct from one another.

The main result of this note is the following theorem. If X is any Hausdorff space with kappa = (F) over cap (X) . (mu) over cap (X) then L( X-(X) is the smallest cardinal phi so that |S| < phi for any set S that is free in X and <(mu)over cap>(X) is the smallest cardinal mu so that, for every set S that is free in X, any open cover of (S) over bar has a subcover of size < mu. Moreover, X-

As defined in [3], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite closed subsets of it intersect. Our main result answers the two main questions of [3] by providing a ZFC construction of a locally countable SAU space of cardinality 2(c) . The construction hinges on the existence of 2(c) weak P-points in omega*, a very deep result of Ken Kunen. It remains open if SAU spaces of cardinality >2(c) could exist, while it was shown in [3] that 2(c) is an upper bound. Also, we do not know if crowded SAU spaces, i.e. ones without any isolated points, exist in ZFC but we obtained the following consistency results concerning such spaces. (1) It is consistent that c is as large as you wish and there is a locally countable and crowded SAU space of cardinality c(+). (2) It is consistent that both c and 2(c) are as large as you wish and there is a crowded SAU space of cardinality 2(c). (3) For any uncountable cardinal kappa the following statements are equivalent: (i) kappa=cof ([kappa](omega), subset of). (ii) There is a locally countable and crowded SAU space of size kappa in the generic extension obtained by adding kappa Cohen reals. (iii) There is a locally countable and countably compact T-1-space of size kappa in some CCC generic extension. (c) 2022 Elsevier B.V. All rights reserved.

It is an interesting, maybe surprising, fact that different dense subspaces of even “nice” topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological space X that we call the double density spectrum of X and denote by dd(X). We improve a result from [1] by showing that dd(X) is always ω-closed (i.e., countably closed) if X is Hausdorff. We manage to give complete characterizations of the double density spectra of Hausdorff and of regular spaces as follows. Let S be a non-empty set of infinite cardinals. Then We also prove a number of consistency results concerning the double density spectra of compact spaces. For instance:

All spaces below are T0 and crowded (i.e. have no isolated points). For n < omega let M(n) be the statement that there are n measurable cardinals, and pi(n) (pi+(n)) that there are n + 1 (0-dimensional T2) spaces whose product is irresolvable. We prove that M(1), pi(1) and pi+(1) are equiconsistent. For 1 < n < omega we show that CON(M(n)) implies CON(pi+(n)). Finally, CON(M(omega)) implies the consistency of having infinitely many crowded 0-dimensional T2 spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malykhin (1973). Concerning an even older question of Ceder and Pearson (1967), we show that the following are consistent modulo a measurable cardinal: (i) There is a 0-dimensional T2 space X with omega 2 < increment (X) < 2 omega 1 whose product with any countable space is not omega 2-resolvable, hence not maximally resolvable. (ii) There is a monotonically normal space X with increment (X) = aleph omega whose product with any countable space is not omega 1-resolvable, hence not maximally resolvable. These significantly improve a result of Eckertson (1997).

For a topological space X we propose to call a subset S subset of X free in X if it admits a well-ordering that turns it into a free sequence in X. The well-known cardinal function F(X) is then definable as sup{vertical bar S vertical bar : S is free in X} and will be called the free set number of X. We prove several new inequalities involving F(X) and F(X-delta), where X-delta is the G(delta)-modification of X: L(X) <= 2(2F(x)) if X is T-2 and L(X) <= 2(F(X)) if X is T-3; vertical bar X vertical bar <= 2(2F(x).)(psi c)((X)) <= 2(2F(X).chi(X)) for any T-2 space X; F(X-delta) <= 2(22F(x)) if X is T-2 and F(X-delta) <= 2(2F(x)) if X is T-3. (C) 2021 The Author(s). Published by Elsevier B.V.

We call a continuous map $f : X \to Y$ nowhere constant if it is not constant on any non-empty open subset of its domain $X$. Clearly, this is equivalent with the assumption that every fiber $f^{-1}(y)$ of $f$ is nowhere dense in $X$. We call the continuous map $f : X \to Y$ pseudo-open if for each nowhere dense $Z \subset Y$ its inverse image $f^{-1}(Z)$ is nowhere dense in $X$. Clearly, if $Y$ is crowded, i.e. has no isolated points, then $f$ is nowhere constant. The aim of this paper is to study the following, admittedly imprecise, question: How small nowhere constant, resp. pseudo-open continuous images can large spaces have? Our main results yield the following two precise answers to this question, explaining also our title. Both of them involve the cardinal function $\widehat{c}(X)$, the hat version of cellularity, which is defined as the smallest cardinal $\kappa$ such that there is no $\kappa$-sized disjoint family of open sets in $X$. Thus, for instance, $\widehat{c}(X) = \omega_1$ means that $X$ is CCC. THEOREM A. Any crowded Tychonov space $X$ has a crowded Tychonov nowhere constant continuous image $Y$ of weight $w(Y) \le \widehat{c}(X)$. Moreover, in this statement $\le$ may be replaced with $<$ iff there are no $\widehat{c}(X)$-Suslin lines (or trees). THEOREM B. Any crowded Tychonov space $X$ has a crowded Tychonov pseudo-open continuous image $Y$ of weight $w(Y) \le 2^{<\widehat{c}(X)}$. If Martin's axiom holds then there is a CCC crowded Tychonov space $X$ such that for any crowded Hausdorff pseudo-open continuous image $Y$ of $X$ we have $w(Y) \ge \mathfrak{c}\,( = 2^{< \omega_1})$.

We call a pair of infinite cardinals $(\kappa,\lambda)$ with $\kappa > \lambda$ a dominating (resp. pinning down) pair for a topological space $X$ if for every subset $A$ of $X$ (resp. family $\mathcal{U}$ of non-empty open sets in $X$) of cardinality $\le \kappa$ there is $B \subset X$ of cardinality $\le \lambda$ such that $A \subset \overline{B}$ (resp. $B \cap U \ne \emptyset$ for each $U \in \mathcal{U}$). Clearly, a dominating pair is also a pinning down pair for $X$. Our definitions generalize the concepts introduced in [GTW] resp. [BT] which focused on pairs of the form $(2^\lambda,\lambda)$. The main aim of this paper is to answer a large number of the numerous problems from [GTW] and [BT] that asked if certain conditions on a space $X$ together with the assumption that $(2^\lambda,\lambda)$ or $((2^\lambda)^+,\lambda)$ is a pinning down pair or \dominating pair for $X$ would imply $d(X) \le \lambda$. [BT] A. Bella, V.V. Tkachuk, Exponential density vs exponential domination, preprint [GTW] G. Gruenhage, V.V. Tkachuk, R.G. Wilson, Domination by small sets versus density, Topology and its Applications 282 (2020)

As it was introduced by Tkachuk and Wilson in [7], a topologicalspace X is cellular-compact if for any cellular, i.e. disjoint, family $$\mathcal{U}$$ of non-emptyopen subsets of X there is a compact subspace $$K \subset X$$ such that $$K \cap U \ne \emptyset$$ foreach $$U \in \mathcal{U}$$. In this note we answer several questions raised in [7] by showing that

The pinning down number pd(X) of a topological space X is the smallest cardinal kappa such that for every neighborhood assignment U on X there is a set of size ? that meets every member of U. Clearly, pd(X) <= d(X) and we call X a pd-example if pd(X) < d(X). We denote by S the class of all singular cardinals that are not strong limit. It was proved in [6] that TFAE: (1) S not equal empty set; (2) there is a 0-dimensional T-2 pd-example; (3) there is a T-2 pd-example. The aim of this paper is to produce pd-examples with further interesting topological properties like connectivity or being a topological group by presenting several constructions that transform given pd-examples into ones with these additional properties. We show that S not equal empty set is also equivalent to the existence of a connected and locally connected T-3 pd-example, as well as to the existence of an abelian T-2 topological group pd-example. However, S not equal empty set in itself is not sufficient to imply the existence of a connected T-3.5 pd-example. But if there is mu is an element of S with mu >= c then there is an abelian T-2 topological group (hence T-3.5) pd-example which is also arcwise connected and locally arcwise connected. Finally, the same assumption S\c not equal empty set even implies that there is a locally convex topological vector space pd-example. (C) 2020 Elsevier B.V. All rights reserved.

The \({G_\delta}\)-modification \({X_\delta}\) of a topological space X is the space on the same underlying set generated by, i.e. having as a basis, the collection of all \({G_\delta}\) subsets of X. Bella and Spadaro recently asked the following question(s): Is \({t(X_\delta) \le 2^{t(X)}}\) true for every (compact) T2 space X?

Given a topological property P, we say that the space X is P-generated if for any subset A subset of X that is not open in X there is a subspace Y subset of X with property P such that A boolean AND Y is not open in Y. (Of course, in this definition we could replace "open" with "closed".) In this paper we prove the following two results: (1) Every Lindelof-generated regular space X satisfying vertical bar X vertical bar = Delta(X) = omega(1) is omega(1)-resolvable. (2) Any (countable extent)-generated regular space X satisfying Delta(X) > omega is omega-resolvable. These are significant strengthenings of our earlier results from [9] which can be obtained from (1) and (2) by simply omitting the "-generated" part. Moreover, the second result improves a recent result of Filatova and Osipov from [4] which states that Lindelof-generated regular spaces of uncountable dispersion character are 2-resolvable. (C) 2019 Elsevier B.V. All rights reserved.

Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or infinite) cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. $X$ contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ is the smallest cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of [van Mill J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016].

A topological space X is called almost discretely Lindelof if every discrete set D subset of X is included in a Lindelof subspace of X. We say that the space X is mu-sequential if for every non-closed set A subset of X there is a sequence of length <= mu in A that converges to a point which is not in A. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. (1) For every almost discretely Lindelof T-3 space X we have vertical bar X vertical bar <= 2(chi(x)). (2) If X is a mu-sequential T-2 space of pseudocharacter psi(X) <= 2(mu) and for every free set D subset of X we have L((D) over bar) <= mu, then vertical bar X vertical bar <= 2(mu). The case chi(X) = omega of (1) provides a solution to Problem 4.5 of [5], while the case mu = omega of (2) is a partial improvement on the main result of [2]. (C) 2018 Published by Elsevier B.V.

Given cardinals ${\lambda}$ and ${\mu}$ we say that $[\mathbf B({\lambda})]^C$ is ${\mu}$-colorable if there is a coloring $f:\mathbf B({\lambda})\to{\mu}$ such that $fZ={\mu}$ whenever a subspace $Z\subset \mathbf B({\lambda})$ is homeomorphic to the Cantor set, where $\mathbf B({\lambda})$ denotes the Baire space of weight ${\lambda}$. We prove that a crowded feebly compact regular space $X$ is ${\mu}$-resolvable provided $[\mathbf B({\lambda})]^C$ is ${\mu}$-colorable for each ${\lambda}<\hat c(X)$. Consequently, (a) every crowded pseudocompact space $X$ with $c(X)< (2^{\omega})^{+{\omega}}$ is $2^{\omega}$-resolvable; (b) if $V=L$, then every crowded pseudocompact space is $2^{\omega}$-resolvable.