Metric spaces with complexity of the smallest infinite ordinal number

In this paper, we are concerned with the study of metric spaces with complexity of the smallest infinite ordinal number. We give equivalent formulations of the definition of metric spaces with complexity of the smallest infinite ordinal number and prove that the exact complexity of the finite product Z≀Z×Z≀Z×⋯×Z≀Z of wreath products is ω, where ω is the smallest infinite ordinal number. Consequently, we obtain that the complexity of (Z≀Z)≀Z is ω+1.