Related papers: Some observations on compact indestructible spaces

A topological space is called {\it dense-separable} if each dense subset of its is separable. Therefore, each dense-separable space is separable. We establish some basic properties of dense-separable topological groups. We prove that each…

For a convergent series with positive terms, we prove that the $\ell^\infty$ product space of bounded subspaces of the Gromov-Hausdorff space can be isometrically embedded into the Gromov-Hausdorff space, where each subspace consists of…

We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations…

We show that it is consistent that for some uncountable cardinal k, all compactifications of the countable discrete space with remainders homeomorphic to $D^k$ are homeomorphic to each other. On the other hand, there are $2^c$ pairwise…

Let $L$ be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of $C_0(L)$ have very strong diameter $2$ properties and, for every real number $\varepsilon$ with $0<\varepsilon<1$, contain an…

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$…

For every couple of Hausdorff functions $ \psi$ and $\varphi $ verifying some mild assumptions, there exists a compact subset $ K $ of the Baire space such that the $ \varphi$-Hausdorff measure and the $ \psi$-packing measure on $ K$ are…

In the present paper we investigate the properties of the Hausdorff mapping $\mathcal{H}$, which takes each compact metric space to the space of its nonempty closed subspaces. It is shown that this mapping is nonexpanding (Lipschitz mapping…

We prove that every fragmentable linearly ordered compact space is almost totally disconnected. This combined with a result of Arvanitakis yields that every linearly ordered quasi Radon-Nikodym compact space is Radon-Nikodym, providing a…

Bonicatto--Pasqualetto--Rajala (2020) proved that a decomposition theorem for sets of finite perimeter into indecomposable sets, known to hold in Euclidean spaces, holds also in complete metric spaces equipped with a doubling measure,…

We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical…

For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a…

In this paper, we show that there are a totally ordered compact K separable (K is Rosenthal compact set), a Hausdorff topology T' on C(K) and two closed subspaces Y1, Y2 of (C(K); Tp) such that (C(K);T') is not universally measurable,…

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are…

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 give examples of $n$-sequentially compact spaces that are not $(n+1)$-sequentially compact under several assumptions. We improve results from Kubis and Szeptycki by building such examples from $\mathfrak{b=c}$ and…

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…

We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space…

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $\mathbf{ZF}$, some are shown to be independent of…

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: For each subset \mathcal{F} of X^{\ast}, which separates X, there exists a countable separating subset \mathcal{F}_{0} of…