Algebraic structures within subsets of Hamel and Sierpiński-Zygmund functions. (English) Zbl 1350.26005

Płotka, Krzysztof

Bull. Belg. Math. Soc. - Simon Stevin 22, No. 3, 447-454 (2015).

The author focuses on specific classes of real-valued functions of one real variable. In particular classes of extendability (Ext) functions, almost continuous (AC) fuctions, Hamel (HF) functions, and Sierpiński-Zygmund (SZ) functions are considered. It is shown that there exists a large semigroup (of cardinality \(2^{\mathfrak{c}}\), where \(\mathfrak{c}\) stands for the cardinality of the set of real numbers) in the intersection of classes HF\(\,\cap\,\)SZ, Ext\(\,\cap\,\)HF, and AC\(\,\cap\,\)HF\(\,\cap\,\)SZ (under some further additional assumptions from set-theory). It is also shown that the lineability of the classs of SZ functions is equal to the lineability of the class of AC\(\,\cap\,\)SZ functions.

Reviewer: Martin Plešinger (Liberec)

MSC:

26A21	Classification of real functions; Baire classification of sets and functions
03E75	Applications of set theory

Keywords:

lineability; almost continuous functions; Hamel functions; Sierpiński-Zygmund functions