This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP2(R,τ(A)) of permutation degree of the Hattori space is discussed. In particular, it is shown that the space SP2(R,τS) contains a closed discrete subset of cardinality c. Moreover, it is shown that the functor SPGn preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces SPGnX and SPGnY are also homotopically equivalent. As a result, it has been proved that the functor SPGn is a covariant homotopy functor.

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In this paper we, prove that if the product Xn of a space X has certain tightness-type properties, then the space of permutation degree SPnX has these properties as well. It is proven that the set tightness (T-tightness) of the space of permutation degree SPnX is equal to the set tightness (T-tightness) of the product Xn.

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In this paper the network-type properties (network,cs−network,cs∗−network,cn−network andck−network) of the spaceSPnGXofG-permutation degree ofXare studied. It is proved that:(1) IfXis aT1-space that has a network of cardinality≤κ, thenSPnGXhas a network of cardinality≤κ;(2) IfXis aT1-space that has acs-network (resp.cs∗-network) ofcardinality≤κ, thenSPnGXhas acs-network (resp.cs∗-network) ofcardinality≤κ;(3) IfXis aT1-space that has acn-network (resp.ck-network) ofcardinality≤κ, thenSPnGXhas acn-network (resp.ck−network) ofcardinality≤κ.