Some results on dimension theory: Universal spaces
A metric space M is said to have the fibered approximation property in dimension n (briefly, M ∈ FAP(n)) if for any ɛ > 0, m ≥ 0 and any map g: m × n → M there exists a map g′: m × n → M such that g′ is ɛ-homotopic to g and dim g′ (z × n) ≤ n for all z ∈ m. The class of spaces having the FAP(n)-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij [11] and Tuncali-Valov [10].
We characterize strong cohomological dimension of separable metric spaces in terms of extension of mappings. Using this characterization, we discuss the relation between strong cohomological dimension and (ordinal) cohomological dimension and give examples to clarify their gaps. We also show that if X is a separable metric ANR and G is a countable Abelian group. Hence for any separable metric ANR X.
Let X and Y be metric compacta such that there exists a continuous open surjection from onto . We prove that if there exists an integer k such that is strongly infinite-dimensional, then there exists an integer p such that is strongly infinite-dimensional.
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.
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.
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.
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 .