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We calculate the cardinal characteristics of the $\sigma$-ideal $\mathscr{H}𝒩\left(G\right)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $cov\left(\mathscr{H}𝒩\right(G\left)\right)\le 𝔟\le max\{𝔡,non(𝒩\left)\right\}\le non\left(\mathscr{H}𝒩\right(G\left)\right)\le cof\left(\mathscr{H}𝒩\right(G\left)\right)>min\{𝔡,non(𝒩\left)\right\}$. If $G={\prod }_{n\ge 0}{G}_n$ is the product of abelian locally compact groups $G_n$, then $add\left(\mathscr{H}𝒩\right(G\left)\right)=add\left(𝒩\right)$, $cov\left(\mathscr{H}𝒩\right(G\left)\right)=min\{𝔟,cov(𝒩\left)\right\}$, $non\left(\mathscr{H}𝒩\right(G\left)\right)=max\{𝔡,non(𝒩\left)\right\}$ and $cof\left(\mathscr{H}𝒩\right(G\left)\right)\ge cof\left(𝒩\right)$, where $𝒩$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $cof\left(\mathscr{H}𝒩\left(G\right)\right)>2^{{\aleph }_0}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of...

It is proven that an infinite-dimensional Banach space (considered as an Abelian topological group) is not topologically isomorphic to a subgroup of a product of $\sigma$-compact (or more generally, $o$-bounded) topological groups. This answers a question of M. Tkachenko.

We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its Stone-Čech compactification $\beta S$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:S\to Y$ to a second countable space $Y$ can be written as the composition $f=g\circ p$ of an open map $p:X\to Z$ onto a second countable space $Z$ and a map $g:Z\to Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.

In each manifold $M$ modeled on a finite or infinite dimensional cube ${[0,1]}^n$, $n\le \omega$, we construct a meager $F_{\sigma }$-subset $X\subset M$ which is universal meager in the sense that for each meager subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h\left(A\right)\subset X$. We also prove that any two universal meager $F_{\sigma }$-sets in $M$ are ambiently homeomorphic.

Using the technique of Fraïssé theory, for every constant $K\ge 1$, we construct a universal object ${𝕌}_K$ in the class of Banach spaces possessing a normalized $K$-suppression unconditional Schauder basis.

We observe that the notion of an almost ${\mathrm{𝔉\Im }}_K$-universal based Banach space, introduced in our earlier paper [1]: Banakh T., Garbulińska-Wegrzyn J., The universal Banach space with a $K$-suppression unconditional basis, Comment. Math. Univ. Carolin. 59 (2018), no. 2, 195–206, is vacuous for $K=1$. Taking into account this discovery, we reformulate Theorem 5.2 from [1] in order to guarantee that the main results of [1] remain valid.

For a functor $F\supset Id$ on the category of metrizable compacta, we introduce a conception of a linear functorial operator $T=\{T_X:Pc\left(X\right)\to Pc\left(FX\right)\}$ extending (for each $X$) pseudometrics from $X$ onto $FX\supset X$ (briefly LFOEP for $F$). The main result states that the functor $S{P}_G^n$ of $G$-symmetric power admits a LFOEP if and only if the action of $G$ on $\{1,\cdots ,n\}$ has a one-point orbit. Since both the hyperspace functor $exp$ and the probability measure functor $P$ contain $S{P}^2$ as a subfunctor, this implies that both $exp$ and $P$ do not admit LFOEP.

A Banach space $X$ is called if for any cover $𝒰$ of $X$ by weakly open sets there is a finite subfamily $𝒱\subset 𝒰$ covering some ball of radius 1 centered at a point $x$ with $\parallel x\parallel \le r$. We prove that an infinite-dimensional separable Banach space $X$ is $\infty$-reflexive ($r$-reflexive for some $r\in \mathbb{N}$) if and only if each $\epsilon$-net for $X$ has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of $X$. We show that the quasireflexive James space $J$ is $r$-reflexive for no $r\in \mathbb{N}$. We do not know if each $\infty$-reflexive...

For a non-isolated point $x$ of a topological space $X$ let ${\mathrm{nw}}_{\chi }\left(x\right)$ be the smallest cardinality of a family $𝒩$ of infinite subsets of $X$ such that each neighborhood $O\left(x\right)\subset X$ of $x$ contains a set $N\in 𝒩$. We prove that (a) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with ${\mathrm{nw}}_{\chi }\left(x\right)={\aleph }_0$; (b) for each point $x\in X$ with ${\mathrm{nw}}_{\chi }\left(x\right)={\aleph }_0$ there is an injective sequence ${\left(x_n\right)}_{n\in \omega }$ in $X$ that $ℱ$-converges to $x$ for some meager filter $ℱ$ on $\omega$; (c) if a functionally Hausdorff space $X$ contains an $ℱ$-convergent injective sequence for some meager filter...

We prove that for an unbounded metric space $X$, the minimal character $𝗆\chi \left(\check{X}\right)$ of a point of the Higson corona $\check{X}$ of $X$ is equal to $𝔲$ if $X$ has asymptotically isolated balls and to $max\{𝔲,𝔡\}$ otherwise. This implies that under $𝔲<𝔡$ a metric space $X$ of bounded geometry is coarsely equivalent to the Cantor macro-cube $2^{<\mathbb{N}}$ if and only if $dim\left(\check{X}\right)=0$ and $𝗆\chi \left(\check{X}\right)=𝔡$. This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.

The Golomb space ${\mathbb{N}}_{\tau }$ is the set $\mathbb{N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb{N}}_{\tau }$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi$ of prime numbers is a dense metrizable subspace of ${\mathbb{N}}_{\tau }$, and each homeomorphism $h$ of ${\mathbb{N}}_{\tau }$ has the following properties: $h\left(1\right)=1$, $h\left(\Pi \right)=\Pi$, ${\Pi }_{h\left(x\right)}=h\left({\Pi }_x\right)$, and $h\left(x^{\mathbb{N}}\right)=h{\left(x\right)}^{\mathbb{N}}$ for all $x\in \mathbb{N}$. Here $x^{\mathbb{N}}:=\{x^n:n\in \mathbb{N}\}$ and ${\Pi }_x$ denotes the set of prime divisors...

The Golomb space ${\mathbb{N}}_{\tau }$ is the set $\mathbb{N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb{N}}_{\tau }$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.