Algebraic properties of the class of Sierpinski-Zygmund functions

by

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].)

• a(SZ) = min{card(F): F is a subset of RR and there is no h in RR s.t. for all f in F function h+f is in SZ}
• m(SZ) = min{card(F): F is a subset of $R$0 and there is no h in RR s.t. for all f in F the product hf is in SZ}
• $c$out(SZ) = min{card(F): F is a subset of $R$1 and there is no h in RR s.t. for all f in F the composition hof is in SZ}
• $c$in(SZ) = min{card(F): F is a subset of $R$2 and there is no h in RR s.t. for all f in F the composition foh is in SZ}
where
• $R$0 = {f in RR: card({x: f(x)=0})< c};
• $R$1 = {f in RR: card({x: f(x)=y})< c for every y in R};
• $R$2 = {f in RR: foh is in SZ for some h in RR}.

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):

• a(SZ)= min{card(F): F subset of RR and for all h in RR there exists f in F s.t. card({x: f(x)=g(x)})=c}.
Moreover, we show that
• m(SZ)=a(SZ);
• if c is a regular cardinal then c< $c$out(SZ)< = $2c$; and
• if c=$k+$ for some cardinal k then $c$out(SZ)=a(SZ);
• $c$in(SZ)=2.
We will also consider "coding" composition numbers $c$r(SZ) and $c$l(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.

Bibliography:

DVI and Postscript files are available at the Topology Atlas preprints side.