The equality of dimensions
The factorization theorem for paracompact -spaces
The homotopy dimension of codiscrete subsets of the 2-sphere 𝕊²
Andreas Zastrow conjectured, and Cannon-Conner-Zastrow proved, that filling one hole in the Sierpiński curve with a disk results in a planar Peano continuum that is not homotopy equivalent to a 1-dimensional set. Zastrow's example is the motivation for this paper, where we characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopy equivalent to 1-dimensional compacta, we prove that each has fundamental group that...
The Interval [0,1] Admits no Functorial Embedding into a Finite-Dimensional or Metrizable Topological Group
An embedding X ⊂ G of a topological space X into a topological group G is called functorial if every homeomorphism of X extends to a continuous group homomorphism of G. It is shown that the interval [0, 1] admits no functorial embedding into a finite-dimensional or metrizable topological group.
The Menger curve Characterization and extension of homeomorphisms of non-locally-separating closed subsets [Book]
The origins of the concept of dimension
The realization of dimension function
The Sorgenfrey plane in dimension theory
The structure of atoms (hereditarily indecomposable continua)
Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting where is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = y ∈ X:ϱ ( x,y) ≤ r coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value...
The structure of compacta satisfying dim(X×X) < 2dimX
Topological games and products, II
Topological type of weakly closed subgroups in Banach spaces
The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in which are interesting from the Banach space theory point of view are discussed.
Transfinite cardinal dimension and separability
Trivial bundles of spaces of probability measures and countable-dimensionality
The probability measure functor P carries open continuous mappings of compact metric spaces into Q-bundles provided Y is countable-dimensional and all fibers are infinite. This answers a question raised by V. Fedorchuk.
Two-dimension-like functions defined on the class of all Tychonoff spaces.
Ultrametric spaces bi-Lipschitz embeddable in
It is proved that if an ultrametric space can be bi-Lipschitz embedded in , then its Assouad dimension is less than n. Together with a result of Luukkainen and Movahedi-Lankarani, where the converse was shown, this gives a characterization in terms of Assouad dimension of the ultrametric spaces which are bi-Lipschitz embeddable in .
Una extensión transfinita de la dimensión por recubrimientos.
Une approche non-standard de la dimension d'un espace topologique
Uniform dimension of mappings (Preliminary communication)
Uniformization of Topological Spaces with Small Inductive Dimension Zero.