Let $\left\{{𝒫}_n\right\}$ be a sequence of covers of a space $X$ such that $\left\{st(x,{𝒫}_n)\right\}$ is a network at $x$ in $X$ for each $x\in X$. For each $n\in \mathbb{N}$, let ${𝒫}_n=\{P_{\beta }:\beta \in {\Lambda }_n\}$ and ${\Lambda }_n$ be endowed the discrete topology. Put $M=\{b=\left({\beta }_n\right)\in {\Pi }_{n\in \mathbb{N}}{\Lambda }_n:\left\{P_{{\beta }_n}\right\}$ forms a network at some point $x_{b}inX\}$ and $f:M⟶X$ by choosing $f\left(b\right)=x_b$ 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 ${𝒫}_n$ is a $c{s}^{*}$-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...

It is shown that every strong $\Sigma$ space is a $D$-space. In particular, it follows that every paracompact $\Sigma$ space is a $D$-space.