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

This paper is dedicated to the memory of Professor Gary Gruenhage (1947–2023). This survey introduces the formation and early development of the topic of point-countable covers and sequence-covering mappings, and lists the recent progress of 38 questions on this topic proposed before 2015, which involve the theory of generalized metric spaces. These questions are related to point-countable covers and sequence-covering mappings, including point-countable covers with certain networks, sequence-covering mappings, images of metric spaces, and hereditarily closure-preserving families.

Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if forevery neighbourhood V of the identity e in G, there exists a natural number n such that G = V ⁿ (resp.we have G = ∪ _n∈N V ⁿ ). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded)premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudoboundedparatopological group such that G is a Lusin space, then is G a topological group. We present some examplesof paratopological groups with interesting properties: (1) There exists a metrizable, zero-dimensional and pseudobounded topological group; (2) There exists a Hausdorff ω-pseudobounded paratopological group G such that G contains a dense subgroupwhich is not ω-pseudobounded; (3) There exists a Hausdorff connected paratopological group which is not ω-pseudobounded.