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It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an $F_\sigma$-subset of a "smaller" dimension. The result applies to different finite or infinite topological dimensions of…

A metric space is plastic if all its non-expansive bijections are isometries. We prove three main results: (1) every countable dense subspace of a normed space is not plastic, (2) every $k$-crowded separable metric space contains a plastic…

A non-trivial separable metric space $X$ is called an almost homology $n$-manifold if the homology groups $H_k(X,X\backslash\{x\},\mathbb Z)$ are trivial for all $x\in X$ and all $k=0,1,..,n-1$. We provide a necessary and sufficient…

A topological space $X$ is $strongly$ $rigid$ if each non-constant continuous map $f:X\to X$ is the identity map of $X$. A Hausdorff topological space $X$ is called $Brown$ if for any nonempty open sets $U,V\subseteq X$ the intersection…

We classify the asymptotic densities of the $\Delta^0_2$ sets according to their level in the Ershov hierarchy. In particular, it is shown that for $n \geq 2$, a real $r \in [0,1]$ is the density of an $n$-c.e.\ set if and only if it is a…

We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved…

We show that it is consistent to have an uncountable sequential group of intermediate sequential order while no countable such groups exist. This is proved by adding $\omega_2$ Cohen reals to a model of $\diamondsuit$.

We prove that the second homology group of the completion of an infinite dimensional free Lie algebra is uncountable.

A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed; the former ad hoc proof of this…

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining…

In what follows, essentially two things will be accomplished: Firstly, it will be proven that a version of the Arzel\`a--Ascoli theorem and the Fr\'echet--Kolmogorov theorem are equivalent to the axiom of countable choice for subsets of…

We show that there is a countable universal abelian p-group for purity, i.e., a countable abelian p-group $U$ such that every countable abelian p-group purely embeds in $U$. This is the last result needed to provide a complete solution to…

This work is devoted to the comparison of de Branges--Rovnyak $H(b)$ spaces harmonically weighted Dirichlet spaces $\mathcal{D}_\mu$. We completely characterize which $H(b)$ spaces are also harmonically weighted Dirichlet spaces…

We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen--Wallach space. We…

We study the reverse mathematics of the theory of countable second-countable topological spaces, with a focus on compactness. We show that the general theory of such spaces works as expected in the subsystem $\mathsf{ACA}_0$ of second-order…

We prove that a compact space is monotonically Sokolov if and only if it is monotonically $\omega$-monolithic. This gives answers to several questions of R. Rojas-Hernandez and V. V. Tkachuk.

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{s,p}(\Omega)$ for an open, bounded set $\Omega\subset\mathbb{R}^{d}$. The density property…

We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are…

We show that it is consistent relative to the existence of suitable large cardinals that for any countable-to-one coloring $c: [\omega_2]^2\to \omega_2$, there exists a closed subset $A\subseteq \omega_2$ of order type $\omega_1$ such that…

Prompted by a recent question of G. Hjorth as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we…