Maximal lineability of the class of Darboux not connectivity maps on R

We provide an elegant argument showing, in ZFC, that the class PES \ Conn of all functions from R to R that are perfectly everywhere surjective (so Darboux) but not connectivity is 2 c -lineable, that is, that there exists a linear subspace of R R of dimension 2 c that is contained in ( PES \ Conn )...

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Bibliographic Details
Published in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2021-10, Vol.115 (4)
Main Author: Ciesielski, Krzysztof Chris
Format: Article
Language:English
Subjects:
Applications of Mathematics
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Original Paper
Theoretical
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Summary:We provide an elegant argument showing, in ZFC, that the class PES \ Conn of all functions from R to R that are perfectly everywhere surjective (so Darboux) but not connectivity is 2 c -lineable, that is, that there exists a linear subspace of R R of dimension 2 c that is contained in ( PES \ Conn ) ∪ { 0 } . This solves a problem from a 2020 paper of Albkwre, Ciesielski, and Wojciechowski. The construction utilizes a transcendental basis of R .
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-021-01103-6