Vol. 94, No. 2, 1981

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ISSN: 0030-8730
𝒦-Borelian embeddings and images of Hausdorff spaces

Brenda MacGibbon

Vol. 94 (1981), No. 2, 369–378
Abstract

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.

Mathematical Subject Classification 2000
Primary: 54C25
Secondary: 54H05
Milestones
Received: 14 January 1979
Published: 1 June 1981
Authors
Brenda MacGibbon