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On three equivalences concerning Ponomarev-systems. (English) Zbl 1164.54363

Summary: Let \(\{\mathcal P_n\}\) be a sequence of covers of a space \(X\) such that \(\{st(x,\mathcal P_n)\}\) is a network at \(x\) in \(X\) for each \(x\in X\). For each \(n\in \mathbb N\), let \(\mathcal P_n=\{P_{\beta }\:\beta \in \Lambda _n\}\) and \(\Lambda _n\) be endowed with the discrete topology. Put \(M=\{b=(\beta _n)\in \Pi _{n\in \mathbb N}\Lambda _ n\: \{P_{\beta _n}\}\) forms a network at some point \(x_b\text{ in }X\}\) and \(f\: M\longrightarrow X\) by choosing \(f(b)=x_b\) for each \(b\in M\). In this paper, we prove that \(f\) is a sequentially-quotient (resp. sequence-covering, compact-covering) mapping if and only if each \(\mathcal P_n\) 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.

MSC:

54E35 Metric spaces, metrizability
54E40 Special maps on metric spaces
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