Internal normality and internal compactness

This text contains an example which presents a way to modify any Dowker space to get a normal space X such that X × [ 0 , 1 ] is not κ-normal, and a theorem implying the existence of a non-Tychonoff space which is internally compact in a larger regular space. It gives answers to several questions by Arhangelʹskii [A.V. Arhangelʹskii, Relative normality and dense subspaces, Topology Appl. 123 (2002) 27–36].