In this paper I will construct a non-separable hereditarily Lindelof space (L space) without any additional axiomatic assumptions. I will also show that there is a function f from [omega_1]^2 to omega_1 such that if A,B, subsets of omega_1, are uncountable and x omega_1, then there are a < b in A and B respectively with f(a,b) = x. Previously it was unknown whether such a function existed even if omega_1 was replaced by 2. Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality aleph_1. Each of these results gives a strong refutation of a well known and longstanding conjecture. The results all stem from the analysis of oscillations of coherent sequences {e_i : i < omega_1} of finite-to-one functions. I expect that the methods presented will have other applications as well.
We introduce a ZFC method that enables us to build spaces (in fact special dense subspaces of certain Cantor cubes) in which we have "full control" over all dense subsets. Using this method we are able to construct, in ZFC, for each uncountable regular cardinal $λ$ a 0-dimensional $T_2$, hence Tychonov, space which is $μ$-resolvable for all $μ< λ$ but not $λ$-resolvable. This yields the final (negative) solution of a celebrated problem of Ceder and Pearson raised in 1967: Are $ω$-resolvable spaces maximally resolvable? This method enables us to solve several other open problems concerning reso…
We show that it is relatively consistent with ZFC that 2^omega is arbitrarily large and every sequence s=(s_i:i<omega_2) of infinite cardinals with s_i<=2^omega is the cardinal sequence of some locally compact scattered space.
We construct locally Lindelöf scattered P-spaces (LLSP spaces, in short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width $ω_1$ and height $ω_2$ and that it is relatively consistent with ZFC that there is an LLSP space of width $ω_1$ and height $ω_3$. Also, we prove a stepping up theorem that, for every cardinal $λ\geq ω_2$, permits us to construct from an LLSP space of width $ω_1$ and height $λ$ satisfying certain additional properties an LLSP space of width $ω_1$ and height $α$ for every ordinal $α< λ^+$. Then, we obt…
We discuss two problems concerning the class Eberlein compacta, i.e., weakly compact subspaces of Banach spaces. The first one deals with preservation of some classes of scattered Eberlein compacta under continuous images. The second one concerns the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. We show that the existence of such Eberlein compacta is consistent with ZFC. We also show that it is consistent with ZFC that each Eberlein compact space of weight $> ω_1$ contains a nonmetrizable closed zero-dimensional subspace…
We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^{aleph_0}<2^{aleph_1}. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size aleph_1, answering a question of the second author.
The topological reconstruction problem asks how much information about a topological space can be recovered from its point-complement subspaces. If the whole space can be recovered in this way, it is called reconstructible. Our main result states that it is independent of the axioms of set theory (ZFC) whether the Stone-Cech remainder of the integers $ω^*$ is reconstructible. Our second result is about the reconstruction of normality. We show that assuming the Continuum Hypothesis, the compact Hausdorff space $ω^*$ has a non-normal reconstruction, namely the space $ω^* \setminus \{p\}$ for a $…
We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if kappa^{<kappa}=kappa then there is such a space of height kappa^+ with only kappa many isolated points. This implies that there is a locally compact scattered space of height omega_2 with omega_1 isolated points in ZFC, solving an old problem of the first author.
We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0 times B_1)= max(irr(B_0),irr(B_1)). We prove consistency of the statement ``there is a Boolean algebra B such that irr(B)<s(B otimes B)'' and we force a superatomic Boolean algebra B_* such that s(B_*)=inc(B_*)=kappa, irr(B_*)=Id(B_*)=kappa^+ and Sub(B_*)=2^(kappa^+). Next we force a superatomic algebra B_0 such that irr(B_0)<inc(B_0) and a superatomic algebra B_1 such that t(B_1)>Aut(B_1). Finally we show that …
The main result of this paper is that, under PFA, for every {\em regular} space $X$ with $F(X) = ω$ we have $|X| \le w(X)^ω$; in particular, $w(X) \le \mathfrak{c}$ implies $|X| \le \mathfrak{c}$. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. We also show that regularity cannot be weakened to Hausdorff in this result because we can find in ZFC a Hausdorff space $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. In fact, this spac…
It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such spaces have countable π-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.
We show that the Continuum Hypothesis is consistent with all regular spaces of hereditarily countable $π$-character being C-closed. This gives us a model of ZFC in which the Continuum Hypothesis holds and compact Hausdorff spaces of countable tightness are sequential.
Modifying a construction of W. Marciszewski we prove (in ZFC) that there exists a subspace of the real line $\mathbb{R}$, such that the realcompact space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology does not admit a continuous bijection onto a $σ$-compact space. This answers a question of Arhangel'skii.
We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question posed by Orihuela, Troyanski and the author in a study of strictly convex norms on Banach spaces. In addition, we show that the Banach space of continuous functions $C_0(D)$ admits a $C^\infty$-smooth bump function.
We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $ω$ implies that the modal logic S4.1.2 is complete with respect to the Čech-Stone compactification of the natural numbers, the space $βω$. In the same fashion we prove that the modal logic S4 is complete with respect to the space $ω^*=βω\setminusω$. This improves the results of G. Bezhanishvili and J. Harding who prove these theorems under stronger assumptions ($\mathfrak{a}=\mathfrak{c}$). Our proof is also somewhat simpler.
It is consistent (relative to ZFC) that the union of max{b,g} many families in the Baire space which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter U, the cofinality of the reduced ultrapower w^w/U is greater than max{b,g}. The model is constructed by oracle chain condition forcing, to which we give a self-contained introduction.
W. Hurewicz proved that analytic Menger sets of reals are $σ$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgöz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We al…
We provide a ZFC example of a compact space K such that C(K)* is w*-separable but its closed unit ball is not w*-separable. All previous examples of such kind had been constructed under CH. We also discuss the measurability of the supremum norm on that C(K) equipped with its weak Baire sigma-algebra.
CONTENTS OF THE ISSUE: Hurewicz-like tests for Borel subsets of the plane; Ordered Spaces, Metric Preimages, and Function Algebras; On the independence of a generalized statement of Egoroff's theorem from ZFC, after T. Weiss; Forty annotated questions about large topological groups; Strong compactness and a partition property; Countable Borel equivalence relations and quotient Borel spaces; Decisive creatures and large continuum; Models of real-valued measurability; Hausdorff ultrafilters; Block combinatorics; Maharam's problem; Universality of uniform Eberlein compacta; Linearly ordered compa…
As defined in [1], 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 [1] by providing a ZFC construction of a locally countable SAU space of cardinality $2^{\mathfrak{c}}$. The construction hinges on the existence of $2^{\mathfrak{c}}$ weak P-points in $ω^*$, a very deep result of Ken Kunen. It remains open if SAU spaces of cardinality $> 2^{\mathfrak{c}}$ could exist, while it was shown in [1] that $2^{2^{\mathfrak{c}}}$ is an upper bound.…
The aim of this paper is to study the Frechet-Urysohn property of the space $Q_p(X,\mathbb{R})$ of real-valued quasicontinuous functions, defined on a Hausdorff space $X$, endowed with the pointwise convergence topology. It is proved that under Suslin's Hypothesis, for an open Whyburn space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable. In particular, it is true in the class of first-countable regular spaces $X$. In ZFC, it is proved that for a metrizable space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable.
In this paper, the class of all linearly ordered topological spaces (LOTS) quasi-ordered by the embeddability relation is investigated. In ZFC it is proved that for countable LOTS this quasi-order has both a maximal (universal) element and a finite basis. For the class of uncountable LOTS of cardinality $κ$ it is proved that this quasi-order has no maximal element for $κ$ at least the size of the continuum and that in fact the dominating number for such quasi-orders is maximal, i.e. $2^κ$. Certain subclasses of LOTS, such as the separable LOTS, are studied with respect to the top and internal …
We answer several questions of V. Tkačuk from [Point-countable $π$-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55--69.] by showing that (1) there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable $π$-base (in fact, the order of any $π$-base of the space is at least $\aleph_ω$); (2) if there is a $κ$-Suslin line then there is a first countable GO space of cardinality $κ^+$ in which the order of any $π$-base is at least $κ$; (3) it is consistent to have a first countable, hereditarily Lindel\" of regular space having uncount…
A space $X$ is called {\it selectively pseudocompact} if for each sequence $(U_{n})_{n\in \mathbb{N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{n})_{n\in \mathbb{N}}$ of points in $X$ such that $cl_X(\{x_n : n < ω\}) \setminus \big(\bigcup_{n < ω}U_n \big) \neq \emptyset$ and $x_{n}\in U_{n}$, for each $n < ω$. Countably compact space spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of $CH$, that for every positive integer $k > 2$ there exists a topological group whose $k$-th power i…
All spaces are assumed to be infinite Hausdorff spaces. We call a space "anti-Urysohn" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that $\bullet$ for every infinite cardinal $κ$ there is a space of size $κ$ in which fewer than $cf(κ)$ many non-empty regular closed sets always intersect; $\bullet$ there is a locally countable AU space of size $κ$ iff $ω\le κ\le 2^{\mathfrak c}$. A space with at least two non-isolated points is called "strongly anti-Urysohn" $($SAU in short$)$ iff any two infinite closed sets in it intersect. We prove that $\bullet$ if $X…
We construct a ZFC example of a nonmetrizable compact space $K$ such that every totally disconnected closed subspace $L\subseteq K$ is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space $K$ must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the the structure of the Banach space $C(K)$, for example, by Holsztyński's theorem, if $K$ is a counterexample, then $C(K)$ contains no isometr…
In this article we will investigate nonmeasurability with respect to some $σ$-ideals in Polish space $X,$ of images of subsets of $X$ by selected mappings defined on the space $X$. Among of them we answer the following question: "It is true that there exists a subset of the unit disc in the real plane such that the continuum many projections onto lines are Lebesgue measurable and continuum many projections are not?". It is known that there exists continuous function $f:[0,1]\to [0,1]$ such that for every Bernstein set $B\subseteq [0,1]$ we have $f[B]=[0,1].$ We show relative consistency with $…
In this issue we celebrate the appearance of the proceedings of the first SPM Workshop, announce several mathematical breakthroughs, have two extended contributions by Babinkostova, and a new open problem by Kalenda. Contents: Editor's note; Proceedings of the first workshop on Coverings, Selections, and Games in Topology; A five element basis for the uncountable linear orders; Set mapping reflection; The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis; A solution to the L space problem and related ZFC constructions; Countable Tightness, Elementary Submodels and Hom…
The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is, in ZFC, a regular (or just Hausdorff) SP of cardinality greater than $\mathfrak c$. While this question is left open, we establish a number of non-trivial results that we list: 1. It is consistent with Martin's Axiom and $\mathfrak c =\aleph_2$ that there is a countably tight and compact SP of cardinality $2^{\mathfrak c}$. 2. If $κ$ is a measurable cardinal then in the forcing extension obtained by…
