Some Properties Of Countably Approximating Posets And Countably C-continuous Posets

The countable S-convergence of nets and generalized countable S-convergence of nets on countably directed complete posets are introduced.Based on them,the characterizations of countably approximating posets by nets and generalized countably approximating posets by nets are given respectively.Then we studied the concept of countable C-continuity on posets.And then,by the lifting method and principal ideal countable C-continuity,we proved that countable C-continuity is equivalent to the principal ideal countable C-continuity on semilattices.