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 In this paper, we prove the semilocality of some multi-anisotropic Sobolev spaces, the density of smooth finite functions in those spaces when Sobolev spaces generated by completely regular Newton polyhedrons and give some examples showing that multi-anisotropic Sobolev space generated by a non-regular Newton polyhedron is not semilocal.

 Newton polyhedron, semilocality of Banach spaces, multi-anisotropic Sobolev spaces.