On $\mathcal{I}$-covering images of metric spaces

Let $\mathcal{I}$ be an ideal on $\mathbb{N}$. A mapping $f:X\to Y$ is called an $\mathcal{I}$-covering mapping provided a sequence $\{y_n{\}}_{n\in \mathbb{N}}$ is $\mathcal{I}$-converging to a point $y$ in $Y$, there is a sequence $\{x_n{\}}_{n\in \mathbb{N}}$ converging to a point $x$ in $X$ such that $x\in f^{-1}(y)$ and each $x_n\in f^{-1}(y_n)$. In this paper we study the spaces with certain $\mathcal{I}$-$cs$-networks and investigate the characterization of the images of metric spaces under certain $\mathcal{I}$-covering mappings, which prompts us to discover $\mathcal{I}$-$csf$-networks. The following main results are obtained: (1) A space $X$ has an $\mathcal{I}$-$csf$-network if and only if $X$ is a continuous and $\mathcal{I}$-covering image of a metric space. (2) A space $X$ is an $\mathcal{I}$-$csf$-countable space if and only if $X$ is a continuous $\mathcal{I}$-covering and boundary $s$-image of a metric space. (3) A space $X$ has a point-countable $\mathcal{I}$-$cs$-network if and only if $X$ is a continuous $\mathcal{I}$-covering and $s$-image of a metric space.