An old question of Arhangel’skii asks if the Menger property of a Tychonoff space X is preserved by homeomorphisms of the space $C_p(X)$ of continuous real-valued functions on X endowed with the pointwise topology. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of function spaces $C_p(X)$ by looking at the way X is positioned in its (Čech–Stone) compactification.

Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $-closed logic ${\mathcal {L}}_\Omega $. ${\mathcal {L}}_\Omega $ is $\omega $-relatively compact iff some $D\in \Omega $ fails to be $\omega _1$-complete iff ${\mathcal {L}}_\Omega $ does not contain the quantifier “there are uncountably many.” If $\Omega $ is a set, or if it contains a countably incomplete ultrafilter, then ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$, if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $, then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact.

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $ , a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $\unicode{x3b3} $ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties.

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter ${\mathcal{F}}$ and prove that it is equivalent to $K({\mathcal{F}})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_{r}(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$, which is not completely metrizable with $P_{r}(X)$ hereditarily Baire. As far as we know, this is the first example of this kind.

We present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s solution to Alexandroff’s problem: A survey, Topology Appl. 153(2006), 2199–2217).

In contrast to previous attempts, we do not need any separation axiom beyond $T_{2}$.

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on ${{\omega }_{1}}$, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ${{\omega }_{1}}$.

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel’s conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.

In this paper we study connections between topological games such as Rothberger, Menger, and compact-open games, and we relate these games to properties involving covers by ${{G}_{\delta }}$ subsets. The results include the following: (1) If TWO has a winning strategy in theMenger game on a regular space $X$, then $X$ is an Alster space. (2) If TWO has a winning strategy in the Rothberger game on a topological space $X$, then the ${{G}_{\delta }}$-topology on $X$ is Lindelöf. (3) The Menger game and the compact-open game are (consistently) not dual.

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space that does not include a perfect pre-image of ${{\text{ }\!\!\omega\!\!\text{ }}_{1}}$ is hereditarily paracompact.

An uncountable cardinal κ is called ${\omega _1}$-strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$-complete ultrafilter on I. We show that the first ${\omega _1}$-strongly compact cardinal, ${\kappa _0}$, cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$. We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$. Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$.

We give easy proofs that (a) the Continuum Hypothesis implies that if the product of $X$ with every Lindelöf space is Lindelöf, then $X$ is a $D$-space, and (b) Borel's Conjecture implies every Rothberger space is Hurewicz.

Extending the work of Larson and Todorcevic, we show that there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form $\text{PFA}\left( S \right)\left[ S \right]$, where $S$ is a coherent Souslin tree.

For any generalized ordered space X with the underlying linearly ordered topological space Xu, let X* be the minimal closed linearly ordered extension of X and be the minimal dense linearly ordered extension of X. The following results are obtained.

1. (1) The projection mapping π:X*→X, π(〈x,i〉)=x, is closed.

2. (2) The projection mapping , ϕ(〈x,i〉)=x, is closed.

3. (3) X* is a monotone D-space if and only if X is a monotone D-space.

4. (4) is a monotone D-space if and only if Xu is a monotone D-space.

5. (5) For the Michael line M, is a paracompact p-space, but not continuously Urysohn.

van Mill et al. posed in ‘Classes defined by stars and neighborhood assignments’, Topology Appl.154 (2007), 2127–2134 the following question: Is a star-compact space metrizable if it has a Gδ-diagonal? In this paper, we give a negative answer to this question.

It is known that the product ${{\omega }_{1}}\times X$ of ${{\omega }_{1}}$ with an ${{M}_{1}}$-space may be non-normal. In this paper we prove that the product $\kappa \times X$ of an uncountable regular cardinal κ with a paracompact semi-stratifiable space is normal iff it is countably paracompact. We also give a sufficient condition under which the product of a normal space with a paracompact space is normal, from which many theorems involving such a product with a countably compact factor can be derived.

In this paper some cardinal inequalities for Urysohn spaces are established. In particular the following two theorems are proved:

(i)If where [A]θ denotes the θ-closed hull of A, i.e., the smallest θ-closed subset of X containing A;

(ii), where aL(X, X) is the smallest cardinal number m such that for every open cover of X there is a subfamily for which

We shall discuss relations between rectangularity and piecewise rectangularity of product spaces. In particular, we show that for each positive integer n there exists an n-dimensional, collectionwise normal, non-piecewise rectangular product X × Y which satisfies the inequality dim (X × Y) ≤ dim X + dim Y.

In [A], Arhangel'skii showed that for any T2 space X, |X|≤2L(x)χ(x), where L(X) is the Lindelöf degree of X and χ(X) is the character of X.

In [B], Bell, Ginsburg and Woods improved this result, assuming normality, by showing that for T4 spaces X, |X|≤2wL(x)χ(x), where wL(X) is the weak Lindelöf degree of X.

We introduce below a new cardinal function aL(X), the almost Lindelöf degree of X, which agrees with L(X) on T3 spaces, but which is often smaller than L(X) on T2 spaces, and show that for T2 spaces X,

A space has the shrinking property if, for every open cover {Va | a ∈ A}, there is an open cover {Wa | a ∈ A} with for each a ∈ A.lt is strangely difficult to find an example of a normal space without the shrinking property. It is proved here that any ∑-product of metric spaces has the shrinking property.

It is shown that in a dense-in-itself Hausdorff space if every set having a dense interior is open, then every compact set is finite.