Classification of metric spaces with infinite asymptotic dimension

Abstract

We introduce a geometric property complementary-finite asymptotic dimension (coasdim). Similar with asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem. Moreover, we show that coasdim(X) ≤ f in + k implies trasdim(X) ≤ ω + k − 1 and transfinite asymptotic dimension of the shift union shIn Section 5, we define the shift union ofiZ is no more than ω + 1. Finally, we give a negative answer to the Question 7.1 raised in [17]. PreliminariesOur terminology concerning the asymptotic dimension follows from [3] and for undefined terminology we refer to [11]. Let (X, d) be a metric space and U, V ⊆ X, let diam U = sup{d(x, y) : x, y ∈ U } and d(U, V ) = inf{d(x, y) : x ∈ U, y ∈ V }. Let R > 0 and U be a family of subsets of X, U is said to be R-bounded if diam U △ = sup{diam U : U ∈ U} ≤ R.U is said to be uniformly bounded if there exists R > 0 such that U is R-bounded. Let r > 0, U is said to be r-disjoint if d(U, V ) ≥ r for every U, V ∈ U and U = V.

Metric spaces with asymptotic property C and finite decomposition complexity

“…Lemma 2.5. (see [9], Proposition 2.1) Let X be a metric space, and let l ∈ N ∪ {0}. Then the following conditions are equivalent:…”

“…Definition 2.6. [9] Every ordinal number γ can be represented as γ = λ (γ) + n(γ), where λ (γ) is the limit ordinal or 0 and n(γ) ∈ N ∪ {0}. Let X be a metric space, we define complementaryfinite asymptotic dimension coasdim(X) inductively as follows:…”

“…Lemma 2.8. (see [9], Theorem 3.3) Let X be a metric space with X 1 , X 2 ⊆ X. Then coasdim(X 1 ∪ X 2 ) ≤ max{coasdim(X 1 ), coasdim(X 2 )}.…”

“…Lemma 2.10. (see [9,12]) Let X and Y be metric spaces, and let ξ be an countable ordinal number. There is a coarse embedding φ : X → Y from X to Y .…”

“…But, if there is a metric space X with ω <trasdim(X) < ∞ was unknown until we constructed a metric space whose transfinite asymptotic dimension is ω + 1 [8]. In [9], we introduced another approach to classify the metric spaces with the infinite asymptotic dimension, which is called the complementary-finite asymptotic dimension (coasdim).…”