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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