A.V. Arhangel'skiı̌ introduced the notion of an almost s-mapping. It is known that the open almost s-images of metric spaces coincide with the open boundary s-images of metric spaces. In this paper, we investigate some questions related to the sequence-covering almost s-images and sequence-covering boundary s-images of metric spaces. We establish some new characterizations of the images of metric spaces under sequence-covering and s-mappings at subsets in topological spaces. The following main results are obtained:

(1) A space X has a $c{s}^{⁎}$-network which is point-countable at non-isolated points if and only if X is a sequentially quotient and almost s-image of a metric space.

(2) A space X has a cs-network which is point-countable at non-isolated points if and only if X is a sequence-covering and almost s-image of a metric space.

(3) A space X has an sn-network which is point-countable at non-isolated points if and only if X is a 1-sequence-covering and almost s-image of a metric space.

(4) A space X is a csf-countable space if and only if X is a sequence-covering (resp., sequentially quotient) and boundary s-image of a metric space.

(5) A space X is an snf-countable space if and only if X is a 1-sequence-covering and boundary s-image of a metric space.

AVArhangel'skiı̌引入了几乎s映射的概念。已知度量空间的开放的几乎s-图像与度量空间的开放边界的s-图像一致。在本文中，我们研究了与度量空间的几乎覆盖s图像的序列和覆盖边界s图像的序列有关的一些问题。我们建立了拓扑空间子集上序列覆盖和s映射下度量空间图像的一些新特征。获得以下主要结果：

（1）空间X具有$C{s}^{⁎}$当且仅当X是度量空间的顺序商且几乎是s映像时，在非隔离点可点计数的网络。

（2）空间X具有cs网络，当且仅当X是度量空间的序列覆盖和几乎s图像时，该网络才能在非孤立点进行点计数。

（3）空间X具有一个sn网络，当且仅当X是度量空间的1序列覆盖和几乎s图像时，该网络才能在非孤立点进行点计数。

（4）当且仅当X是度量空间的序列覆盖（分别为商数）和边界s图像时，空间X是csf可计数的空间。

（5）当且仅当X是度量空间的1序列覆盖和边界s图像时，空间X是snf可数空间。