We introduce and investigate inductive dimensions 𝒦 -Ind and ℒ-Ind for classes 𝒦 of finite simplicial complexes and classes ℒ of ANR-compacta (if 𝒦 consists of the 0-sphere only, then the 𝒦 -Ind dimension is identical with the classical large inductive dimension Ind). We compare K-Ind to K-Ind introduced by the author [Mat. Vesnik 61 (2009)]. In particular, for every complex K such that K * K is non-contractible, we construct a compact Hausdorff space X with K-Ind X not equal to K-dim X.

Let X be a compactum and let $A=(A_i,B_i):i=1,2,...$ be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed $F_i$ separating $A_i$ and $B_i$ the intersection $\left(\cap F_i\right)\cap Y$ is not empty. So A is inessential on Y if there exist closed $F_i$ separating $A_i$ and $B_i$ such that $\cap F_i$ does not intersect Y. Properties of inessentiality are studied and applied to prove:  Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on which A is...

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{\alpha }$, $Y_{\alpha }$ and $Z_{\alpha }$ such that (i) $f{X}_{\alpha },f{Y}_{\alpha },f{Z}_{\alpha }=\omega ₀$, where f is either trdef or ₀-trsur, (ii) $A\left(\alpha \right)-trind{X}_{\alpha }=\infty$ and $M\left(\alpha \right)-trind{X}_{\alpha }=-1$, (iii) $A\left(\alpha \right)-trind{Y}_{\alpha }=-1$ and $M\left(\alpha \right)-trind{Y}_{\alpha }=\infty$, and (iv) $A\left(\alpha \right)-trind{Z}_{\alpha }=M\left(\alpha \right)-trind{Z}_{\alpha }=\infty$ and $A(\alpha +1)\cap M(\alpha +1)-trind{Z}_{\alpha }=-1$. We also show that there exists no separable metrizable space $W_{\alpha }$ with $A\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$, $M\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$ and $A\left(\alpha \right)\cap M\left(\alpha \right)-trind{W}_{\alpha }=\infty$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

We prove that the set of all Krasinkiewicz maps from a compact metric space to a polyhedron (or a 1-dimensional locally connected continuum, or an n-dimensional Menger manifold, n ≥ 1) is a dense $G_{\delta }$-subset of the space of all maps. We also investigate the existence of surjective Krasinkiewicz maps from continua to polyhedra.

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 X be a compact Hausdorff space and M a metric space. $E_0(X,M)$ is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which $E_0(X,M)$ is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of $E_0(X,M)$ as a normed linear space. We also build three first countable Eberlein...