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In this paper, we study the space
of
-permutation degree. Especially, new properties of this space are investigated. We study its metrazibility, semi-metrazibility, stratifiability, semi-stratifiability and uniformizability. Moreover, we study the uniform space
investigating when this space is totally bounded (
-bounded). Finally, a study of universal elements in classes of spaces that are presented in this article, and related open questions complete this paper.