Let $𝒟$ denote a true dimension function, i.e., a dimension function such that $𝒟\left({ℝ}^n\right)=n$ for all $n$. For a space $X$, we denote the hyperspace consisting of all compact connected, non-empty subsets by $C\left(X\right)$. If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence ${\left({𝒟}_{\ge n}\left(C\left(X\right)\right)\right)}_{n=2}^{\infty }$ is $F_{\sigma }$-absorbing in $C\left(X\right)$. As a consequence, there is a homeomorphism $h:C\left(X\right)\to Q^{\infty }$ such that for all $n$, $h\left[\{A\in C\left(X\right):𝒟\left(A\right)\ge n+1\}\right]=B^n\times Q\times Q\times \cdots$, where $B$ denotes the pseudo boundary of the Hilbert cube $Q$. It follows that if $X$ is a countable infinite product of non-degenerate...

The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that $X\tau K(H_k\left(L\right),k)$ for all k ≥ 1, then $X\tau K({\pi }_k\left(F\right),k)$ and $X\tau K({\pi }_k\left(L\right),k)$ for all k ≥ $.$Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈ ANR. Suppose...

Soit $f:U\to V$ une application analytique propre entre des ouverts de ${\mathbf{C}}^n$, soit $X$ un sous-ensemble analytique de $U$ et soit $Z=f^{-1}\left(f\right(X\left)\right)$. On donne des conditions pour que $\overline{Z-X}\cap X$ soit de codimension 1 dans $X$.

In accordance with the Bing-Borsuk conjecture, we show that if X is an n-dimensional homogeneous metric ANR continuum and x ∈ X, then there is a local basis at x consisting of connected open sets U such that the cohomological properties of Ū and bd U are similar to the properties of the closed ball ⁿ ⊂ ℝⁿ and its boundary ${}^{n-1}$. We also prove that a metric ANR compactum X of dimension n is dimensionally full-valued if and only if the group Hₙ(X,X∖x;ℤ) is not trivial for some x ∈ X. This implies that...

Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{\sigma }$-subset $A_k$ of X such that $dim{A}_k\le k$ and the restriction $f\left|\right(X\setminus A_k)$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about...

A topological space X is called an ${\check{H}}^n$-bubble (n is a natural number, ${\check{H}}^n$ is Čech cohomology with integer coefficients) if its n-dimensional cohomology ${\check{H}}^n\left(X\right)$ is nontrivial and the n-dimensional cohomology of every proper subspace is trivial. The main results of our paper are: (1) Any compact metrizable ${\check{H}}^n$-bubble is locally connected; (2) There exists a 2-dimensional 2-acyclic compact metrizable ANR which does not contain any ${\check{H}}^2$-bubbles; and (3) Every n-acyclic finite-dimensional $L{\check{H}}^n$-trivial metrizable compactum...

We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.