Related papers: Countable dense homogeneity in powers of zero-dime…
Our first motivation was the question: can a countable structure have an automorphism group, which a free uncountable group? This is answered negatively in [Sh:744]. Lecturing in a conference in Rutgers, February 2001, I was asked whether I…
In this paper, we introduced $\alpha$-Hurewicz $\&$ $\theta$-Hurewicz properties in a topological space $X$ and investigated their relationship with other selective covering properties. We have shown that for an extremally disconnected…
We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space X=(X,T) is countably compact iff it is countably subcompact relative…
It is shown that if $G$ is an uncountable Polish group and $A\subseteq G$ is a universally measurable set such that $A^{-1}A$ is meager, then the set $T_l(A)=\{\mu\in P(G): \mu(gA)=0 \text{for all} g\in G\}$ is co-meager. In particular, if…
Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic…
We investigate the behavior of functional countability and exponential separability in products and subspaces of topological spaces. We solve a problem of Tkachuk by showing that the product of functionally countable pseudocompact spaces is…
We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.
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…
We show (in ZFC) that the cardinality of a compact homogeneous space of countable tightness is no more than the size of the continuum.
Let X be an uncountable Polish space. Lubica Hola showed recently that there are 2^continuum many quasi-continuous real valued functions defined on the uncountable Polish space that are not Borel measurable. Inspired by Hola's result, we…
This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) contains an open subgroup H with ample…
For a metrizable space $X$ and a finite measure space $(\Omega,\mathfrak{M},\mu)$ let $M_{\mu}(X)$ and $M^f_{\mu}(X)$ be the spaces of all equivalence classes (under the relation of equality almost everywhere mod $\mu$) of…
We show that a set of non-negative reals is the distance set of a separable complete metric space if and only if it is either countable or is an analytic set which has 0 as a limit point. We also consider spaces with simpler distance sets.
A topological space $X$ is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product $X=\prod X_i$ of almost discrete spaces $X_i$ the space $C_p(X)$ of continuous real-valued…
Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpi\'nski and Knaster-Urbanik theorems are presented. Embeddable properties of countable metric spaces are generalized onto uncountable metric…
We prove that there exists a $\Sigma^0_1$ closed subset of $[0,1]$ that is not homeomorphic to any computably compact space. We show that the index set of c.e. subspaces of $[0,1]$ that admit a computably compact presentation is not…
We use countable metric spaces to code Polish metric spaces and evaluate the complexity of some statements about these codes and of some relations that can be determined by the codes. Also, we propose a coding for continuous functions…
All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows…
A Hausdorff topological space $X$ is called $\textit{superconnected}$ (resp. $\textit{coregular}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\bar U_1\cap\dots\cap\bar U_n$ is not empty…
We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…