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.

Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power $X^{\omega }$ of any subspace X ⊂ Y is not universal for the class ₂ of absolute $G_{\delta \sigma }$-sets; moreover, if Y is an absolute $F_{\sigma \delta }$-set, then $X^{\omega }$ contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute $G_{\delta }$-set, then $X^{\omega }$ contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power $X^{\omega }$ of...