In this note we study the relation between $k_R$-spaces and $k$-spaces and prove that a $k_R$-space with a $\sigma$-hereditarily closure-preserving $k$-network consisting of compact subsets is a $k$-space, and that a $k_R$-space with a point-countable $k$-network consisting of compact subsets need not be a $k$-space.

The concepts of $k$-systems, $k$-networks and $k$-covers were defined by A. Arhangel’skiǐ in 1964, P. O’Meara in 1971 and R. McCoy, I. Ntantu in 1985, respectively. In this paper the relationships among $k$-systems, $k$-networks and $k$-covers are further discussed and are established by $mk$-systems. As applications, some new characterizations of quotients or closed images of locally compact metric spaces are given by means of $mk$-systems.

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...