The Alexandroff one-point compactification as a prototype for extensions

Using the Alexandroff one-point compactification as a point of departure, we study a general procedure for building an extension 〈X∪I,τ0〉〈X∪I,τ0〉 of a topological space 〈X,τ〉〈X,τ〉, given a family {ℬi:i∈I}{ℬi:i∈I} of nontrivial closed ideals on XX, indexed by the intended remainder II.