Related papers: Notes on linearly H-closed spaces and od-selection…

A space is od-compact (resp. od-Lindel\"of) provided any cover by open dense sets has a finite (resp. countable) subcover. We first show with simple examples that these properties behave quite poorly under finite or countable unions. We…

We use topological consequences of PFA, MA$_{\omega_1}$(S)[S] and PFA(S)[S] proved by other authors to show that normal first countable linearly H-closed spaces with various additionals properties are compact in these models.

We generalized the characterization of H-closedness for linearly ordered pospaces as follows: A pospace $X$ without an infinite antichain is an H-closed pospace if and only if $X$ is a directed complete and down-complete poset such that sup…

We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are…

Answering a question raised by V. V. Tkachuk, we present several examples of $\sigma$-compact spaces, some only consistent and some in ZFC, that are not countably tight but in which the closure of any discrete subset is countably tight. In…

Let $\mathcal{F} $ be a pointwise almost periodic decomposition of a compact metrizable space $X$. Then $\mathcal{F} $ is $R$-closed if and only if $\hat{\mathcal{F}} $ is usc. Moreover, if there is a finite index normal subgroup $H$ of an…

A space $X$ is od-Menger if it satisfies $\mathsf{U_{fin}}(\Delta_X, \mathcal{O}_X)$, where $\mathcal{O}_X,\Delta_X$ are the collection of covers of $X$ by respectively open subsets and open dense subsets. We show that under CH, there is a…

We prove that: 1. If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable. 2. A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally…

We show that if a separable space X has a meager open subset containing a copy of the Cantor set 2^\omega, then X has $\frak{c}$ types of countable dense subsets. We suggest a generalization of the \lambda-set for non-separable spaces. Let…

Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \in \{\omega D,…

The property of being selectively separable is well-studied and generalizations such as H-separable and wH-separable have also generated much interest. Bardyla, Maesano, and Zdomskyy proved from Martin's Axiom that there are countable…

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued…

In this paper we study the behaviour of selective separability properties in the class of Frech\'{e}t-Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Frech\'{e}t-Urysohn (hence…

Suppose G is a topological group containing a (closed) topological copy of the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal k-space) then every closed metrizable subset in $G$ is locally compact. Applying this…

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of…

A space has $\sigma$-compact tightness if the closures of $\sigma$-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable…

An open chain cover $\{U_\alpha : \alpha\in\kappa\}$ ($\kappa$ a cardinal) of a space $X$ is a systematic cover if the closure of $U_\alpha$ is contained in $U_\beta$ when $\alpha<\beta$, and $X$ is Type I if $\kappa=\omega_1$ and the…

We show that all sufficiently nice $\lambda$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $\kappa \le \mathfrak{b}$ there is a countable dense homogeneous metric space…

A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove…

We prove that a closed subgroup $H$ of a second countable locally compact group $G$ is amenable if and only if its left regular representation on an Orlicz space $L^\Phi(G)$ for some $\Delta_2$-regular $N$-function $\Phi$ almost has…