## On three equivalences concerning Ponomarev-systems

*Ying** Ge*

**Address****.**

Department of Mathematics, Suzhou
University, Suzhou
215006, P. R. China

**E-mail. ** geying@pub.sz.jsinfo.net

**Abstract****.** 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 the discrete
topology. Put $M=\{b=(\beta_n)\in\Pi_{n\in{\mathbb N}}\Lambda_ n: \{P_{\beta_n}\}$ {\it forms a network at some point\/} $x_b\ in \ X\}$ 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.

**AMSclassification**.
54E35, 54E40.

**Keywords****. **Ponomarev-system, point-star
network, $cs^*$-(resp. $fcs$-,
$cfp$-)cover, sequentially-quotient (resp. sequence-covering, compact-covering) mapping** .**