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G δ Covers and Large Thin Sets of Reals
Author(s) -
Fremlin D. H.,
Jasiński J.
Publication year - 1986
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-53.3.518
Subject(s) - mathematics , regular cardinal , continuum hypothesis , combinatorics , hausdorff space , cover (algebra) , cardinal number (linguistics) , axiom , discrete mathematics , geometry , mathematical analysis , mechanical engineering , linguistics , philosophy , engineering
A strong G δ ‐ covering number is the cardinal of some minimal cover of a compact Hausdorff space by G δ sets (2B). We show that every cardinal below the first weakly inaccessible cardinal is a strong G δ ‐covering number (2F), but that certain types of large cardinal are not (2J, 2K). If Martin's axiom is true, a cardinal less than the continuum is a strong G δ ‐covering number if and only if it is the cardinal of a minimal cover of R by Borel sets (4F). On the other hand, it appears to be consistent with Martin's axiom that there should be a cardinal less than the continuum which is not a strong G δ ‐covering number, and in the same context there are subsets of R , of cardinal the continuum, which are ‘thin’ by any of a very large number of criteria (§ 5). We end by giving an application of the same ideas to the theory of Borel‐dense sets (§ 6).