The following two questions
are discussed in this article: (1) under what conditions is a Hausdorff space
embeddable as a 𝒦-Borelian subset of some Hausdorff space; and (2) under what
conditions is the Hausdorff 1 − 1 continuous image of a 𝒦-Borelian subset of a
Hausdorff space a 𝒦-Borelian subset of some Hausdorff space containing it. We
obtain necessary and sufficient conditions in answer to question (1) in the
case of a 𝒦σδ and necessary conditions for 𝒦-Borelian subsets of each class
α if the containing space is a perfectly σ-normal 𝒦σ. Question 2 does not
always have a positive answer as is shown by an example of a Hausdorff 1 − 1
continuous image of 𝒦-Borelian subset of a compact Hausdorff space which is not
𝒦-Borelian in any Hausdorff space containing it. In partial answer to question
(2) necessary conditions on the domain and the range of the function are
presented.
