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In this paper, the local density (ld) and the local weak density (lwd) in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree SP and the subfunctor of permutation degree SP G , P is the cardinal number of topological spaces. Let X be an infinite T1-space. We prove that the following propositions hold. (1) Let Y n ⊂ X; (A) if d (Y ) = d (X), then d (SPY ) = d (SPX); (B) if lwd (Y ) = lwd (X), then lwd (SPY ) = lwd (SPX). (2) Let Y ⊂ X; (A) if ld (Y ) = ld (X), then ld (SPY ) = ld (SPX); (B) if wd (Y ) = wd (X), then wd (SPY ) = wd (SPX). (3) Let n be a positive integer, and let G be a subgroup of the permutation group Sn. If X is a locally compact T1-space, then SPX, SP GX, and expn X are k-spaces. (4) Let n be a positive integer, and let G be a subgroup of the permutation group Sn. If X is an infinite T1-space, then nπw (X) = nπ w (SPX) = nπw (

THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE