APD profiles and transfinite asymptotic dimension

We develop the theory of APD profiles introduced by J. Dydak for infinity-pseudometric spaces ([3]). We connect them with transfinite asymptotic dimension defined by T. Radul ([4]). We give a characterization of spaces with transfinite asymptotic dimension at most omega + n for n is an element of omega and a sufficient condition for a space to have transfinite asymptotic dimension at most m center dot<bold> </bold>omega + n for m, n is an element of omega, using the language of APD profiles. This condition together with a result from [7] enables us to answer positively the question in [3, 5.15]. (C) 2020 Elsevier B.V. All rights reserved.