Networks in Ponomarev-systems

Abstract

Let \(f, M, X, P\) be a Ponomarev-system. We prove that \(f\) is a compact-covering mapping if \(P\) is a strong \(k\)-network of \(X\) Furthermore, \(f\) is a compact-covering, \(s\)-mapping if \(P\) is a point-countable \(c/p\) -network (or point-countable strong \(k\) network) of \(X\). As an application of these results, a point-countable cover of a space is a strong \(k\)-network if it is a \(c/p\) -network, where `point-countable' can not be omitted.