On a core concept of Arhangelʹskiĭ

Arhangelʹskiĭ [A.V. Arhangelʹskiĭ, Locally compact spaces of countable core and Alexandroff compactification, Topology Appl. 154 (2007) 625–634] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper.