Mappings with 1-dimensional absolute neighborhood retract fibers
We prove that if f: X → Y is a closed surjective map between metric spaces such that every fiber belongs to a class S of spaces, then there exists an -set A ⊂ X such that A ∈ S and for all y ∈ Y. Here, S can be one of the following classes: (i) M: e-dim M ≤ K for some CW-complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = M: dim M ≤ n, then dim f ∆ g ≤ 0 for almost all .
We prove that an ultrametric space can be bi-Lipschitz embedded in if its metric dimension in Assouad’s sense is smaller than n. We also characterize ultrametric spaces up to bi-Lipschitz homeomorphism as dense subspaces of ultrametric inverse limits of certain inverse sequences of discrete spaces.
We study retracts of coset spaces. We prove that in certain spaces the set of points that are contained in a component of dimension less than or equal to n, is a closed set. Using our techniques we are able to provide new examples of homogeneous spaces that are not coset spaces. We provide an example of a compact homogeneous space which is not a coset space. We further provide an example of a compact metrizable space which is a retract of a homogeneous compact space, but which is not a retract of...
We give a proof of a theorem of Maćkowiak on the existence of universal n-dimensional hereditarily indecomposable continua, based on the Baire-category method.
Let . Then cmp Zₙ < def Zₙ for n ≥ 5. This is the answer to a question posed by de Groot and Nishiura [GN] for n ≥ 5.
We characterize Peano continua using Bing-Krasinkiewicz-Lelek maps. Also we deal with some topics on Whitney preserving maps.