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We consider uniformly continuous surjections between $C_p(X)$ and $C_p(Y)$ (resp, $C_p^*(X)$ and $C_p^*(Y$)) and show that if $X$ has some dimensional-like properties, then so does $Y$. In particular, we prove that if $T:C_p(X)\to C_p(Y)$ is a continuous linear surjection, then $\dim Y=0$ if $\dim X=0$. This provides a positive answer to a question raised by Kawamura-Leiderman \cite[Problem 3.1]{kl}., Comment: 24 pages

This is a survey of the recent results and unsolved problems about locally compact homogeneous metric spaces. Mostly, homogeneous finite-dimensional $ANR$-spaces are discussed., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:2003.06907

We prove that any region $\Gamma$ in a homogeneous $n$-dimensional and locally compact separable metric space $X$, where $n\geq 2$, cannot be irreducibly separated by a closed $(n-1)$-dimensional subset $C$ with the following property: $C$ is acyclic in dimension $n-1$ and there is a point $b\in C\cap\Gamma$ having a special local base $\mathcal B_C^b$ in $C$ such that the boundary of each $U\in\mathcal B_C^b$ is acyclic in dimension $n-2$. In case $X$ is strongly locally homogeneous, it suffices to have a point $b\in C\cap\Gamma$ with an ordinary base $\mathcal B_C^b$ satisfying the above condition. The acyclicity means triviality of the corresponding \v{C}ech cohomology groups. This implies all known results concerning the separation of regions in homogeneous connected locally compact spaces., Comment: 10 pages

We investigate to what extend finite-dimensional homogeneous locally compact $ANR$-spaces have common properties with Euclidean manifolds. Specially, the local structure of homogeneous $ANR$-spaces is described. Using that description, we provide a positive solution of the problem whether every finite-dimensional homogeneous metric $ANR$-compactum $X$ is dimensionally full-valued, i.e. $\dim X\times Y=\dim X+\dim Y$ for any metric compactum $Y$., Comment: 19 pages

We describe the proper absolute (neighborhood) extensors for the class of at most $n$-dimensional spaces, notation $\rm{A(N)E}_p(n)$. For example, the unique locally compact $n$-dimensional separable metric space $X\in\rm{AE}_p(n)$ satisfiyng the $\rm{DD^nP}$-property is the $n$-dimensional Menger compactum without a point. Non-metrizable $\rm{A(N)E}_p(n)$-spaces are also described., Comment: 9 pages

It was shown by van Mill and Valov that regions in strongly locally homogeneous locally compact metric spaces of dimension $\ge 2$ are not separated by arcs. We improve this result by replacing strong local homogeneity with homogeneity. Moreover, we prove the result for the case when only one end point of an arc is in the interior of the region.

We investigate to what extend the density of $Z_n$-maps in the characterization of $Q$-manifolds, and the density of maps $f\in C(\mathbb N\times Q,X)$ having discrete images in the $l_2$-manifolds characterization can be weakened to the density of homological $Z_n$-maps and homological $Z$-maps, respectively. As a result, we obtain homological characterizations of $Q$-manifolds and $l_2$-manifolds., Comment: 15 pages

This is a survey of most important results and unsolved problems about homogeneous finite-dimensional metric $ANR$-compacta. We also discuss some partial results and possible ways of solutions., Comment: 15 pages

The aim of this paper is to investigate the class of quasi $\kappa$-metrizable spaces. This class is invariant with respect to arbitrary products and contains Shchepin's $\kappa$-metrizable spaces as a proper subclass., Comment: 13 pages

Near-openly generated groups are introduced. It is a topological and multiplicative subclass of $\mathbb R$-factorizable groups. Dense and open subgroups, quotients and Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, lindel\" off almost metrizable groups and the spaces $C_p(X)$ of all continuous real-valued functions on a Tychonoff space $X$ with pointwise convergence topology are near-openly generated. We provide characterizations of near-openly generated groups using methods of inverse spectra and topological game theory.

We prove that if $X$ is a strongly locally homogeneous and locally compact separable metric space and $G$ is a region in $X$ with $\dim G=2$, then $G$ is not separated by any arc in $G$., Comment: 7 pages

We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups., Comment: 12 pages

The aim of this paper is to extend the external characterization of I-favorable spaces. This allows us to obtain a characterization of compact I-favorable spaces in terms of quasi k-metrics. We also provide proofs of some author's results announced in [14]., Comment: 16 pages. arXiv admin note: text overlap with arXiv:1005.0074

We provide some properties and characterizations of homologically $UV^n$-maps and $lc^n_G$-spaces. We show that there is a parallel between recently introduced by Cauty algebraic $ANR$'s and homologically $lc^n_G$-metric spaces, and this parallel is similar to the parallel between ordinary $ANR$'s and $LC^n$-metric spaces. We also show that there is a similarity between the properties of $LC^n$-spaces and $lc^n_G$-spaces. Some open questions are raised., Comment: 20 pages

There are different definitions of homological dimension of metric compacta involving either \v{C}ech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to different groups. It is shown that all homological dimensions of a metric compactum X with respect to any field coincide provided X is homologically locally connected with respect to the singular homology up to dimension n=dim X. We also prove that any two-dimensional lc^2 metric compactum X satisfies the equality dim(X times Y)=dim X+dim Y for any metric compactum Y. This improves the well known result of Kodama that every two-dimensional ANR is dimensionally full-valued. Actually, the condition X to be lc^2 can be weaken to the existence at every point x a neighborhood V of x such that the inclusion homomorphism H_k(V;S^1)\to H_k(X;S^1)$ is trivial for all k=1,2., Comment: 10 pages

We introduce the notion of set-wise injective maps and provide results about fiber embeddings. Our results improve some previous results in this area., Comment: 11 pages

The homological dimension $d_G$ of metric compacta was introduced by Alexandroff. In this paper we provide some general properties of $d_G$, mainly with an eye towards describing the dimensional full-valuedness of compact metric spaces. As a corollary of the established properties of $d_G$, we prove that any two-dimensional $lc^2$ metric compactum is dimensionally full-valued. This improves the well known result of Kodama that every two-dimensional $ANR$ is dimensionally full-valued. Applications for homogeneous metric $ANR$-compacta are also given., Comment: 18 pages

In accordance with the Bing-Borsuk conjecture \cite{bb}, we show that if $X$ is an $n$-dimensional homogeneous metric $ANR$ compactum and $x\in X$, then there is a local basis at x consisting of connected open sets U such that the homological properties of \bar U and bdU are similar to the properties of the closed ball B^n in R^n and its boundary S^{n-1}. We discuss also the following questions raised by Bing-Borsuk, where X is a homogeneous ANR-compactum with dim X=n: Is it true that $X$ is cyclic in dimension n? Is it true that no non-empty closed subset of X, acyclic in dimension n-$, separates X? It is shown that both questions have positive answers simultaneously, and a positive solution to each one of them implies a solution to another question of Bing-Borsuk (whether every finite-dimensional homogenous metric AR-compactum is a point).

A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically UV^n set-valued maps., Comment: Journal of Fixed Point Theory and Applications

Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club collections, consisting of countable families of co-zero sets. For example, a Tychonoff space $X$ is skeletally Dugundji if and only if there exists an additive $c$-club on $X$. Dugundji spaces are characterized by the existence of additive $d$-clubs.

In accordance with the Bing-Borsuk conjecture, we show that if X is an n-dimensional homogeneous metric ANR compactum and x\in X, then there is a local basis at x consisting of connected open sets U such that the cohomological properties of \overline U and bdU are similar to the properties of the closed ball \mathbb B^n\subset\mathbb R^n and its boundary \mathbb S^{n-1}. We also prove that a metric ANR compactum X of dimension n is dimensionally full-valued if and only if the group H_n(X,X\setminus x) is not trivial for some x\in X. This implies that every 3-dimensional homogeneous metric ANR compactum is dimensionally full-valued., Comment: 12 pages

We provide a machinery for transferring some properties of metrizable $ANR$-spaces to metrizable $LC^n$-spaces. As a result, we show that for complete metrizable spaces the properties $ALC^n$, $LC^n$ and $WLC^n$ coincide to each other. We also provide the following spectral characterizations of $ALC^n$ and cell-like compacta: A compactum $X$ is $ALC^n$ if and only if $X$ is the limit space of a $\sigma$-complete inverse system $S=\{X_\alpha, p^{\beta}_\alpha, \alpha<\beta<\tau\}$ consisting of compact metrizable $LC^n$-spaces $X_\alpha$ such that all bonding projections $p^{\beta}_\alpha$, as a well all limit projections $p_\alpha$, are $UV^n$-maps. A compactum $X$ is a cell-like (resp., $UV^n$) space if and only if $X$ is the limit space of a $\sigma$-complete inverse system consisting of cell-like (resp., $UV^n$) metric compacta., Comment: 16 pages

The following selection theorem is established:\\ Let $X$ be a compactum possessing a binary normal subbase $\mathcal S$ for its closed subsets. Then every set-valued $\mathcal S$-continuous map $\Phi\colon Z\to X$ with closed $\mathcal S$-convex values, where $Z$ is an arbitrary space, has a continuous single-valued selection. More generally, if $A\subset Z$ is closed and any map from $A$ to $X$ is continuously extendable to a map from $Z$ to $X$, then every selection for $\Phi|A$ can be extended to a selection for $\Phi$. This theorem implies that if $X$ is a $\kappa$-metrizable (resp., $\kappa$-metrizable and connected) compactum with a normal binary closed subbase $\mathcal S$, then every open $\mathcal S$-convex surjection $f\colon X\to Y$ is a zero-soft (resp., soft) map. Our results provide some generalizations and specifications of Ivanov's results (see \cite{i1}, \cite{i2}, \cite{i3}) concerning superextensions of $\kappa$-metrizable compacta., Comment: 8 pages

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq 0$ is a $V^n$-continuum in the sense of Alexandroff (1957). We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu (2007) whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.

Arvanitakis established recently a theorem which is a common generalization of Michael's convex selection theorem and Dugundji's extension theorem. In this note we provide a short proof of a more general version of Arvanitakis' result., Comment: 8 pages

We describe the supports of a class of real-valued maps on $C*(X)$ introduced by Radul. Using this description, a characterization of compact-valued retracts of a given space in terms of functional extenders is obtained. For example, if $X\subset Y$, then there exists a continuous compact-valued retraction from $Y$ onto $X$ if and only if there exists a normed weakly additive extender $u\colon C*(X)\to C*(Y)$ with compact supports preserving $\min$ (resp., $\max$) and weakly preserving $\max$ (resp., $\min$). Similar characterizations are obtained for upper (resp., lower) semi-continuous compact-valued retractions. These results provide characterizations of (not necessarily compact) absolute extensors for zero-dimensional spaces, as well as absolute extensors for one-dimensional spaces, involving non-linear functional extenders., Comment: 19 pages

Using a factorization theorem due to Pasynkov we provide a short proof of the existence and density of parametric Bing and Krasinkiewicz maps. In particular, the following corollary is established: Let $f\colon X\to Y$ be a surjective map between paracompact spaces such that all fibers $f^{-1}(y)$, $y\in Y$, are compact and there exists a map $g\colon X\to\mathbb I^{\aleph_0}$ embedding each $f^{-1}(y)$ into $\mathbb I^{\aleph_0}$. Then for every $n\geq 1$ the space $C^*(X,\mathbb R^n)$ of all bounded continuous functions with the uniform convergence topology contains a dense set of maps $g$ such that any restriction $g|f^{-1}(y)$, $y\in Y$, is a Bing and Krasinkiewicz map., Comment: 5 pages

We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_\sigma$-set $A\subset X$ such that $A\in\mathrm S$ and $\dim f^{-1}(y)\backslash A=0$ for all $y\in Y$. Here, $\mathrm S$ can be one of the following classes: (i) $\{M:\mathrm{e-dim}M\leq K\}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm S=\{M:\dim M\leq n\}$, then $\dim f\triangle g\leq 0$ for almost all $g\in C(X,\mathbb I^{n+1})$., Comment: 10 pages

We provide both a spectral and an internal characterizations of arbitrary I-favorable spaces with respect to co-zero sets. As a corollary we establish that any product of compact I-favorable spaces with respect to co-zero sets is also I-favorable with respect to co-zero sets. We also prove that every C*-embedded I-favorable with respect to co-zero sets subspace of an extremally disconnected space is extremally disconnected., Comment: 17 pages

A metric space $M$ us said to have the fibered approximation property in dimension $n$ (br., $M\in \mathrm{FAP}(n)$) if for any $\epsilon>0$, $m\geq 0$ and any map $g: I^m\times I^n\to M$ there exists a map $g':I^m\times I^n\to M$ such that $g'$ is $\epsilon$-homotopic to $g$ and $\dim g'\big(\{z\}\times I^n\big)\leq n$ for all $z\in I^m$. The class of spaces having the $\mathrm{FAP}(n)$-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij and Tuncali-Valov.

The book is devoted to constructing embedding finite-dimensional maps into trivial bundles and investigating the corresponding general position properties., Comment: This is a survey of principal results of a future book

A characterization of $\varkappa$-metrizable compacta in terms of extension of functions and usco retractions into superextensions is established., Comment: 12 pages

The notion of a regular operator with compact supports between function spaces is introduced. On that base we obtain a characterization of absolute extensors for zero-dimensional spaces in terms of regular extension operators having compact supports. Milyutin maps are also considered and it is established that some topological properties, like paracompactness, metrizability and k-metrizability, are preserved under Milyutin maps., Comment: 26 pages

Let $M$ be a complete metric $ANR$-space such that for any metric compactum $K$ the function space $C(K,M)$ contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that $M$ has the following property: If $f\colon X\to Y$ is a perfect surjection between metric spaces, then $C(X,M)$ with the source limitation topology contains a dense $G_\delta$-subset of maps $g$ such that all restrictions $g|f^{-1}(y)$, $y\in Y$, are Bing (resp., Krasinkiewicz) maps. We apply the above result to establish some mapping theorems for extensional dimension., Comment: 12 pages

We say that a metrizable space $M$ is a Krasinkiewicz space if any map from a metrizable compactum $X$ into $M$ can be approximated by Krasinkiewicz maps (a map $g\colon X\to M$ is Krasinkiewicz provided every continuum in $X$ is either contained in a fiber of $g$ or contains a component of a fiber of $g$). In this paper we establish the following property of Krasinkiewicz spaces: Let $f\colon X\to Y$ be a perfect map between metrizable spaces and $M$ a Krasinkiewicz complete $ANR$-space. If $Y$ is a countable union of closed finite-dimensional subsets, then the function space $C(X,M)$ with the source limitation topology contains a dense $G_{\delta}$-subset of maps $g$ such that all restrictions $g|f^{-1}(y)$, $y\in Y$, are Krasinkiewicz maps. The same conclusion remains true if $M$ is homeomorphic to a closed convex subset of a Banach space and $X$ is a $C$-space., Comment: 14 pages

The class of metrizable spaces $M$ with the following approximation property is introduced and investigated: $M\in AP(n,0)$ if for every $\e>0$ and a map $g\colon\I^n\to M$ there exists a 0-dimensional map $g'\colon\I^n\to M$ which is $\e$-homotopic to $g$. It is shown that this class has very nice properties. For example, if $M_i\in AP(n_i,0)$, $i=1,2$, then $M_1\times M_2\in AP(n_1+n_2,0)$. Moreover, $M\in AP(n,0)$ if and only if each point of $M$ has a local base of neighborhoods $U$ with $U\in AP(n,0)$. Using the properties of AP(n,0)-spaces, we generalize some results of Levin and Kato-Matsuhashi concerning the existence of residual sets of $n$-dimensional Lelek maps., Comment: 34 pages

Every open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed subsets of X so that their image by f is Y provided all fibers of f are infinite and C*-embedded in X. Applications are demonstrated for the existence of "disjoint" usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and for special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions., Comment: 16 pages

We establish cohomological and extension dimension versions of the Hurewicz dimension-raising theorem, Comment: 19 pages

Here are two of our main results: Theorem 1. Let X be a normal space with dim X=n and m\geq n+1. Then the space C*(X,R^m) of all bounded maps from X into R^m equipped with the uniform convergence topology contains a dense G_{\delta}-subset consisting of maps g such that \bar{g(X)}\cap\Pi^d is at most (n+d-m)-dimensional for every d-dimensional plane \Pi^d in R^m, where m-n\leq d\leq m. Theorem 2. Let X be a metrizable compactum with dim X\leq n and m\geq n+1. Then, C(X,R^m) contains a dense G_{\delta}-subset of maps g such that for any integers t,d,T with 0\leq t\leq d\leq m-n-1 and d\leq T\leq m and any d-plane \Pi^d\subset R^m parallel to some coordinate planes \Pi^t\subset\Pi^T in R^m, the inverse image g^{-1}(\Pi^d) has at most q points, where q=d+1-t+\frac{n+(n+T-m)(d-t)}{m-n-d} if n\geq (m-n-T)(d-t) and q=1+\frac{n}{m-n-T} otherwise. In case m=2n+1, the combination of Theorem 1 and the Nobeling--Pontryagin embedding theorem provides a generalization of a theorem due to Roberts. Theorem 2 extends the following results: the N\"{o}beling--Pontryagin embedding theorem (d=0, m=T\geq 2n+1); Hurewicz's theorem about mappings into an Euclidean space with preimages of small cardinality (d=0, n+1\leq m=T\leq 2n); Boltyanski's theorem about k-regular maps (d=k-1, t=0, T=m\geq nk+n+k) and Goodsell's theorem about existence of special embeddings (t=0, T=m). An infinite-dimensional analogue of Theorem 2 is also established. Our results are based on Theorem 1.1 below which is considered as a converse assertion of the transversal Tverberg's theorem and implies the Berkowitz-Roy theorem., Comment: 21 pages

Let L be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension not greater than L contains a universal element which is an absolute extensor in dimension L. Our main result shows that L is quasi-finite.

We extend the definition of quasi-finite complexes by considering not necessarily countable complexes. We provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Further, we extend the definition of UV(L)-spaces on non-compact case and show that some properties of UV(n)-spaces and UV(n)-maps remain valid, respectively, for UV(L)-spaces and UV(L)-maps., Comment: 21 pages

We prove that any closed map between metrizable spaces can be extended to a closed map between completely metrizable spaces with the same extensional dimension., Comment: 6 pages

Let $f\colon X\to Y$ be a perfect map between finite-dimensional metrizable spaces and $p\geq 1$. It is shown that the space $C^*(X,\R^p)$ of all bounded maps from $X$ into $\R^p$ with the source limitation topology contains a dense $G_{\delta}$-subset consisting of $f$-regularly branched maps. Here, a map $g\colon X\to\R^p$ is $f$-regularly branched if, for every $n\geq 1$, the dimension of the set $\{z\in Y\times\R^p: |(f\times g)^{-1}(z)|\geq n\}$ is $\leq n\cdot\big(\dim f+\dim Y\big)-(n-1)\cdot\big(p+\dim Y\big)$. This is a parametric version of the Hurewicz theorem on regularly branched maps., Comment: 12 pages

Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that, for every positive integer $p\geq 1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$ with the source limitation topology such that if $g\in{\mathcal H}(k,m,p)$, then each fiber of $f\triangle g$ contains at most $\max\{m+k-p+2,1\}$ points.This result provides a proof of two hypotheses of S. Bogatyi, V. Fedorchuk and J. van Mill., Comment: 8 pages

Let $f\colon X\to Y$ be a perfect surjective map of metrizable spaces. It is shown that if $Y$ is a $C$-space (resp., $\dim Y\leq n$ and $\dim f\leq m$), then the function space $C(X,\uin^{\infty})$ (resp., $C(X,\uin^{2n+1+m})$) equipped with the source limitation topology contains a dense $G_{\delta}$-set $\mathcal{H}$ such that $f\times g$ embeds $X$ into $Y\times\uin^{\infty}$ (resp., into $Y\times\uin^{2n+1+m}$) for every $g\in\mathcal{H}$. Some applications of this result are also given., Comment: 11 pages

Characterizations of paracompact finite $C$-spaces via continuous selections are given. We apply these results to obtain some properties of finite $C$-spaces. Factorization theorems and a completion theorem for finite $C$- spaces are also established., Comment: 13 pages, to appear in Set-Valued Analysis