arXiv subjects

V. Valov

Publications and source records attributed to V. Valov.

At least 19 recordsLinked to original sources

arXiv2024

On uniformly continuous surjections between $C_p$-spaces over metrizable spaces

Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T: C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of all real-valued continuous (resp., continuous and bounded) functions on $X$ endowed with the pointwise convergence topology. We show that if additionally $T$ is an inversely bounded mapping and $X$ has some dimensional-like property $\mathcal P$, then so does $Y$. For example, this is true if $\mathcal P$ is one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. Also, we consider other properties $\mathcal P$: of being a scattered, or a strongly $\sigma$-scattered space, or being a $\Delta_1$-space (see [17]). Our results strengthen and extend several results from [6], [13], [17].

math.GN↗

arXiv2023

On generalized $V^n$-continua

The notion of a $V^n$-continuum was introduced by Alexandroff \cite{ps} as a generalization of the concept of $n$-manifold. In this note we consider the cohomological analogue of $V^n$-continuum and prove that any strongly locally homogeneous generalized continuum $X$ with cohomological dimension $\dim_G X=n$ is a generalized $V^n$-space with respect to the cohomological dimension $\dim_G$. In particular, every strongly locally homogeneous continuum of covering dimension $n$ is a $V^n$-continuum in the sense of Alexandroff. This provides a partial answer to a question raised in \cite{tv}. An analog of the Mazurkiewicz theorem that no subset of covering dimension $\le n-2$ cuts any region of the Euclidean $n$-space is also obtained for strongly locally homogeneous generalized continua $X$ of cohomological dimension $\dim_G X=n$.

math.GN↗

arXiv2022

On homogeneity of $\mathbb N^\tau$

It is shown that any homeomorphism between two compact subsets of $\mathbb N^\tau$ can be extended to an autohomeomorphism of $\mathbb N^\tau$.

math.GN↗

arXiv2021

Extending homeomorphisms on Cantor cubes

We discuss the question of extending homeomorphism between closed subsets of the Cantor discontinuum $D^\tau$. It is established that any homeomorphism $f$ between two closed subsets of $D^\tau$ can be extended to an autohomeomorphism of $D^\tau$ provided $f$ preserves the $\lambda$-interiors of the sets for every cardinal $\lambda$.

math.GN↗

arXiv2021

On homogeneity of Cantor cubes

It is established that any homeomorphism between two closed negligible subset of $D^\tau$ can be extended to an autohomeomorphism of $D^\tau$.

math.GN↗

arXiv2020

On $Q$-manifolds bundles

We prove a homological characterization of $Q$-manifolds bundles over $C$-spaces. This provides a partial answer to Question QM22 from \cite{w}.

math.GT↗

arXiv2014

Alexandroff type manifolds and homology manifolds

We introduce and investigate the notion of (strong) $K^n_G$-manifolds, where $G$ is an abelian group. One of the result related to that notion (Theorem 3.4) implies the following partial answer to the Bing-Borsuk problem \cite{bb}, whether any partition of a homogeneous metric $ANR$-space $X$ of dimension $n$ is cyclic in dimension $n-1$: If $X$ is a homogeneous metric $ANR$ compactum with $\check{H}^{n}(X;G)\neq 0$, then $\check{H}^{n-1}(M;G)\neq 0$ for every set $M\subset X$, which is cutting $X$ between two disjoint open subsets of $X$. Another implication of Theorem 3.4 (Corollary 3.6) provides an analog of the classical result of Mazurkiewicz \cite{ma} that no region in $\mathbb R^n$ can be cut by a subset of dimension $\leq n-2$. Concerning homology manifolds, it is shown that if $X$ is arcwise connected complete metric space which is either a homology $n$-manifold over a group $G$ or a product of at least $n$ metric spaces, then $X$ is a Mazurkiewicz arc $n$-manifold. We also introduce a property which guarantees that $H_k(X,X\setminus x;G)=0$ for every $x\in X$ and $k\leq n-1$, where $X$ is a homogeneous locally compact metric $ANR$.

math.GN↗

arXiv2014

Skeletally generated spaces and absolutes

Some properties of skelatally generated spaces are established. In particular, it is shown that any compactum co-absolute to a $κ$-metrizable compactum is skeletally generated. We also prove that a compactum $X$ is skeletally generated if and only if its superextension $λX$ is skeletally Dugundji and raise some natural questions.

math.GN↗

arXiv2014

Homogeneous ANR-spaces and Alexandroff manifolds

We specify a result of Yokoi \cite{yo} by proving that if $G$ is an abelian group and $X$ is a homogeneous metric $ANR$ compactum with $\dim_GX=n$ and $\check{H}^n(X;G)\neq 0$, then $X$ is an $(n,G)$-bubble. This implies that any such space $X$ has the following properties: $\check{H}^{n-1}(A;G)\neq 0$ for every closed separator $A$ of $X$, and $X$ is an Alexandroff manifold with respect to the class $D^{n-2}_G$ of all spaces of dimension $\dim_G\leq n-2$. We also prove that if $X$ is a homogeneous metric continuum with $\check{H}^n(X;G)\neq 0$, then $\check{H}^{n-1}(C;G)\neq 0$ for any partition $C$ of $X$ such that $\dim_GC\leq n-1$. The last provides a partial answer to a question of Kallipoliti and Papasoglu \cite{kp}.

math.GT↗

arXiv2012

Skeletally Dugundji spaces

We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. The main result states that the following conditions are equivalent for a given space $X$: (i) $X$ is skeletally Dugundji; (ii) Every compactification of $X$ is co-absolute to a Dugundji space; (iii) Every $C^*$-embedding of the absolute $p(X)$ in another space is strongly $π$-regular; (iv) $X$ has a multiplicative lattice in the sense of Shchepin \cite{s76} consisting of skeletal maps.

math.GN↗

arXiv2012

Generalized Cantor manifolds and indecomposable continua

We review results concerning homogeneous compacta and discuss some open questions. It is established that indecomposable continua are Alexandroff (resp., Mazurkiewicz, or strong Cantor) manifolds with respect to the class of all continua. We also provide some new proofs of Bing's theorems about separating metric compacta by hereditarily indecomposable compacta.

math.GN↗

arXiv2010

Very I-favorable spaces

We prove that a Hausdorff space $X$ is very $\mathrm I$-favorable if and only if $X$ is the almost limit space of a $σ$-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open bonding maps. It is also shown that the class of Tychonoff very $\mathrm I$-favorable spaces with respect to the co-zero sets coincides with the d-openly generated spaces.

math.GN↗

arXiv2010

Embeddings of finite-dimensional compacta in Euclidean spaces

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $q$ is an integer, let $B_{q,d,m}(g)$ be the set of all lines $Π^d\subset\mathbb R^m$ such that $|g^{-1}(Π^d)|\geq q$. Let also $\mathcal H(q,d,m,k)$ denote the maps $g\colon X\to\mathbb R^m$ such that $\dim B_{q,d,m}(g)\leq k$. We prove that for any $n$-dimensional metric compactum $X$ each of the sets $\mathcal H(3,1,m,3n+1-m)$ and $\mathcal H(2,1,m,2n)$ is dense and $G_δ$ in the function space $C(X,\mathbb R^m)$ provided $m\geq 2n+1$ (in this case $\mathcal H(3,1,m,3n+1-m)$ and $\mathcal H(2,1,m,2n)$ can consist of embeddings). The same is true for the sets $\mathcal H(1,d,m,n+d(m-d))\subset C(X,\mathbb R^m)$ if $m\geq n+d$, and $\mathcal H(4,1,3,0)\subset C(X,\mathbb R^3)$ if $\dim X\leq 1$.

math.GN↗

arXiv2010

Special embeddings of finite-dimensional compacta in Euclidean spaces

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $z\not\in g(X)$, let $P_{2,1,m}(g,z)$ be the set of all lines $Π^1\subset\mathbb R^m$ containing $z$ such that $|g^{-1}(Π^1)|\geq 2$. We prove that for any $n$-dimensional metric compactum $X$ the functions $g\colon X\to\mathbb R^m$, where $m\geq 2n+1$, with $\dim P_{2,1,m}(g,z)\leq 0$ for all $z\not\in g(X)$ form a dense $G_δ$-subset of the function space $C(X,\mathbb R^m)$. A parametric version of the above theorem is also provided.

math.GN↗

arXiv2010

Sections, Selections and Prohorov's Theorem

The famous Prohorov theorem for Radon probability measures is generalized in terms of usco mappings. In the case of completely metrizable spaces this is achieved by applying a classical Michael result on the existence of usco selections for l.s.c. mappings. A similar approach works when sieve-complete spaces are considered.

math.GN↗

arXiv2009

Mazurkiewicz manifolds and homogeneity

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

math.GN↗

arXiv2008

Probability measures and Milyutin maps between metric spaces

We prove that the functor $\Hat{P}$ of Radon probability measures transforms any open map between completely metrizable spaces into a soft map. This result is applied to establish some properties of Milyutin maps between completely metrizable spaces.

math.GN↗

arXiv2008

Generalized Cantor manifolds and homogeneity

A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds, and $V^n$-continua, and prove corresponding versions of the above theorem. We apply our results to show that each homogeneous metrizable continuum which is not in a given class $\mathcal C$ is a strong Cantor manifold (or at least a Cantor manifold) with respect to $\mathcal C$. Here, the class $\mathcal C$ is one of four classes that are defined in terms of dimension-like invariants. A class of spaces having bases of neighborhoods satisfying certain special conditions is also considered.

math.GN↗