Characterization theorem for the anti-Cantor set
by
Ihor Zarichnyi
Ivan Franko National University of Lviv

Let Z₂ ^{< w} be the direct sum ⊕_{i ∈ w}Z₂ of the two-element groups Z₂={0, 1}, endowed with the metric d((x_i), (y_i))=max{2ⁱ|x_i-y_i|:i ∈ w}. The metric space Z₂ ^{< w} called the anti-Cantor set is an asymptotic counterpart of the Cantor cube Z₂^w endowed with the metric d((x_i), (y_i))=max{2^-i|x-y_i|:i ∈ w}.

A multi-valued map F:X  ⇒ Y between two metric spaces is called bornologous if ∀d > 0 ∃e > 0 ∀A ⊂ X [diam(A) < d  ⇒  diam(F(A)) < e].

Two metric spaces X, Y are called coarsely equivalent if there is a multi-valued map F:X⇒ Y such that F(X)=Y, F^-1(Y)=X and F, F^-1 both are bornologous.

Characterization theorem for anti-Cantor set. A metric space (Y, r) is coarsely equivalent to the anti-Cantor set if and only if there exist a > 0, n ∈ N, there exist strongly increasing sequences (a_i)_{i ∈ N} and (n_i)_{i ∈ N} of natural and real numbers respectively that tend to infinity such that for every i the set Y can be represented as disjoint union of a countable family of sets {Y_j}_{j ∈ N} such that, for all j, k ∈ N, diam (Y_j) ≤ a_i, dist(Y_j, Y_k) > a_i-1, and every set Y_j can be covered by at least 2^n_i-n and at most 2^n_i sets of diameter ≤ a.

Using this characterization theorem one can prove that the n-th hypersymmetric power exp_n(Z₂ ^{< w}) of the anti-Cantor set is coarsely equivalent to Z₂ ^{< w}.

Date received: May 9, 2008