The compactificability classes of certain spaces

By a space, we always mean topological space. Throughout the paper, we mostly use the standard topological notions as in [1] or [3] with the exception that all spaces are assumed without any separation axioms. Especially, compactness is understood without the Hausdorff separation axiom. Some definitions (with broad references and explanations) of less standard notions (related especially to non-Hausdorff topology) may be found in the recent book [4]. The reader may find some topological notions (usually also non-Hausdorff) related to computer science and logic in [13] as well as in [4]. We take the terminology related to θ-regularity from [5, 7], but a relevant source is also [4]. An ordinal number is taken to be the set of smaller ordinals, and a cardinal number is the smallest ordinal equipotent with some fixed set. Let S be a set. We denote the cardinality of S by |S|. Let (X ,τ) be a space. For our convenience and simplicity, sometimes we will speak just about the space X , while meaning, more precisely, the pair (X ,τ). Similarly, if we first start speaking about the space X without specifying its topology explicitly, later we will usually denote the topology of X by τ or τX (in the case that we will work simultaneously with more topological spaces or more topologies on the same set). The weight of a space (X ,τ) is defined as the least infinite cardinal number w(X) such that (X ,τ) has an open base τ0 ⊆ τ with |τ0| ≤ w(X). The spaces with w(X) = א0 are called second countable. In a space X , a point x ∈ X is in the θ-closure of a set A⊆ X (x ∈ clθ A) if every