We continue the research on lineability/spaceability properties in sequence spaces. Our main results show that several theorems this framework are valid a stricter sense. To do job we introduce notion of pointwise lineability which seems to be independent interest for further investigations other environments.

We characterize sequence-covering (resp., 1-sequence-covering, 2-sequence-covering) mssc-images of locally separable metric spaces by means of σ-locally finite cs-networks (resp., sn-networks, so-networks) consisting of א0-spaces (resp., sn-second countable spaces, so-second countable spaces). As the applications, we get characterizations of certain sequence-covering, quotient mssc-images of lo...