Algebraic properties of the class of Sierpinski-Zygmund functions


Krzysztof Ciesielski & Tomasz Natkaniec

28 pages; Topology Appl. 79 (1997), 75--99.

We define and examine cardinal invariants connected with algebraic operations on Sierpinski-Zygmund functions.

Recall that a function f: R-->R is of Sierpinski-Zygmund type (shortly, an SZ function) if the restriction of f to M is discontinuous for any set subset M of R with cardinality card(M) equal to continuum c, the cardinality of R.

We study the following cardinals, where RR stands for the class of all functions from R to R. (Compare [CM], [CR], [Na] and [NR].)


We prove that c< a(SZ) < = 2c and a(SZ) can be equal to any regular cardinal between c+ and 2c. (In particular, each f in RR can be expressed as the sum of two SZ functions.) Moreover, we compare a(SZ) with a(Darboux), and give the following combinatorial characterization of a(SZ):

Moreover, we show that We will also consider "coding" composition numbers cr(SZ) and cl(SZ) defined in a similar way and notice that it is consistent that they are equal to 1, while it is also consistent that they are "big."

In our considerations we use generalized Martin's Axiom and Lusin sequence axiom.


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