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Ponomarev-system; point-star network; $cs^*$-(resp. $fcs$-; $cfp$-)cover; sequentially-quotient (resp. sequence-covering; compact-covering) mapping
Let $\lbrace {\mathcal P}_n\rbrace $ be a sequence of covers of a space $X$ such that $\lbrace st(x,{\mathcal P}_n)\rbrace $ is a network at $x$ in $X$ for each $x\in X$. For each $n\in \mathbb N$, let ${\mathcal P}_n=\lbrace P_{\beta }:\beta \in \Lambda _n\rbrace $ and $\Lambda _ n$ be endowed the discrete topology. Put $M=\lbrace b=(\beta _n)\in \Pi _{n\in \mathbb N}\Lambda _ n: \lbrace P_{\beta _n}\rbrace $ forms a network at some point $x_b\ in \ X\rbrace $ 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.
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