On Three Equivalences concerning Ponomarev-systems

Let {Pn} be a sequence of covers of a space X such that {st(x,Pn)} is a network at x inX for each x ∈ X. For each n ∈ N, let Pn = {Pβ : β ∈ Λn} and Λn be endowed the discrete topology. Put M = {b = (βn) ∈ Πn∈NΛn : {Pβn} forms a network at some point xb in X} and f : M −→ X by choosing f(b) = xb for each b ∈ M . In this paper, we prove that f is a sequentiallyquotient (resp. sequence-covering, compact-covering) mapping if and only if each Pn is a cs-cover (resp. fcs-cover, cfp-cover) of X. As a consequence of this result, we prove that f is a sequentially-quotient, s-mapping if and only if it is a sequence-covering, s-mapping, where “s” can not be omitted.