Glasnik matematički, Vol. 47 No. 2, 2012.
Original scientific paper
https://doi.org/10.3336/gm.47.2.17
Property A and asymptotic dimension
Matija Cencelj
; Pedagoška fakulteta, Univerza v Ljubljani, Kardeljeva pl. 16, SI-1111 Ljubljana, Slovenija
Jerzy Dydak
orcid.org/0000-0003-3302-9881
; University of Tennessee, Knoxville, TN 37996, USA
Aleš Vavpetič
; Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska ulica 19, SI-1111 Ljubljana, Slovenija
Abstract
The purpose of this note is to characterize the asymptotic dimension asdim(X) of metric spaces X in terms similar to Property A of Guoliang Yu. We prove that for a metric space (X,d) and n≥ 0 the following conditions are equivalent:
asdim(X,d)≤ n.
For each R,ε > 0 there is S > 0 and finite non-empty subsets Ax⊂ B(x,S)× N, x X, such that |AxΔ Ay| / |Ax∩ Ay| < ε if d(x,y) < R and the projection of Ax onto X contains at most n+1 elements for all x X.
For each R > 0 there is S > 0 and finite non-empty subsets Ax⊂ B(x,S)× N, x X, such that |AxΔ Ay| / |Ax∩ Ay| < 1/(n+1) if d(x,y) < R and the projection of Ax onto X contains at most n+1 elements for all x X.
Keywords
Asymptotic dimension; Property A
Hrčak ID:
93960
URI
Publication date:
19.12.2012.
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