APD profiles and transfinite asymptotic dimension

We develop the theory of APD profiles introduced by J. Dydak for ∞-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 ω + n for n ∈ ω and a sufficient condition for a space to have transfinite asymptotic dimension at most m ⋅ ω + n for m , n ∈ ω , 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] .