Remainders in extensions and finite unions of locally compact sets

We discuss the relationship between properties of spaces and their remainders in extensions from the class P _fin of all finite unions of locally compact spaces. In particular, we show that a space X ∈ P _fin iff the remainder in each (some) compactification of X is in P _fin. Then we study the class P _fin and the relationship between the remainders of a space from this class in compact extensions and give a generalization of the theorem of Henriksen-Isbell.