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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...
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...
We prove a Z-set unknotting theorem for Nöbeling spaces.
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.
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.
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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