Some network-type properties of the space of G-permutation degree

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