Rational acyclic resolutions.
Cell-like resolutions preserving cohomological dimensions.
Extensions of maps to the projective plane.
Bing maps and finite-dimensional maps
Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open. Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map such that dim (f × g) = 1. We improve this result of Sternfeld showing...
Inessentiality with respect to subspaces
Let X be a compactum and let 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 separating and the intersection is not empty. So A is inessential on Y if there exist closed separating and such that 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...
A Z-set unknotting theorem for Nöbeling spaces
We prove a Z-set unknotting theorem for Nöbeling spaces.
A dimensional property of Cartesian product
We show that the Cartesian product of three hereditarily infinite-dimensional compact metric spaces is never hereditarily infinite-dimensional. It is quite surprising that the proof of this fact (and this is the only proof known to the author) essentially relies on algebraic topology.
Universal acyclic resolutions for arbitrary coefficient groups
We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective -map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that we have and r is G-acyclic.
Hyperspaces of two-dimensional continua
Let X be a compact metric space and let C(X) denote the space of subcontinua of X with the Hausdorff metric. It is proved that every two-dimensional continuum X contains, for every n ≥ 1, a one-dimensional subcontinuum with . This implies that X contains a compact one-dimensional subset T with dim C (T) = ∞.
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