The box-counting dimension of the sine-curve
The covering dimension of product of generalized Sorgenfrey lines
The dimension of analytic sets
The dimension of hyperspaces of non-metrizable continua
We prove that, for any Hausdorff continuum X, if dim X ≥ 2 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then either dim C(X) = 2 or C(X) is not a C-space. This generalizes some results known for metric continua.
The dimension of metrizable subspaces of Eberlein compacta and Eberlein compactifications of metrizable spaces
We prove that every Baire subspace Y of c₀(Γ) has a dense metrizable subspace X with dim X ≤ dim Y. We also prove that the Kimura-Morishita Eberlein compactifications of metrizable spaces preserve large inductive dimension. The proofs rely on new and old results concerning the dimension of uniform spaces.
The dimension of products of complete separable metric spaces
The dimension of remainders of rim-compact spaces
Answering a question of Isbell we show that there exists a rim-compact space X such that every compactification Y of X has dim(Y)≥ 1.
The Dimension of the Set of All Finite Subsets of the Continuum
The dimension of X^n where X is a separable metric space
For a separable metric space X, we consider possibilities for the sequence where . In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is such that , , for n >1, such that , and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of is shown to exist which satisfies and .
The Dimension Print of Most Convex Surfaces.
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