### On Ciesielski's Problems

by Shelah. [Sh:675]

J Applied Analysis, 1997

We discuss some problems posed by Ciesielski. For
example we show that, consistently, d_c is a singular cardinal and
e_c<d_c . Next we prove that the Martin Axiom for
sigma --centered forcing notions implies that for every function
f:R^2 ---> R there are functions
g_n,h_n:R ---> R, n< omega, such that
f(x,y)= sum_{n=0}^{infty} g_n(x)h_n(y) . Finally, we deal with
countably continuous functions and we show that in the Cohen model
they are exactly the functions f with the property that
(for all U in [R]^{aleph_1})(exists U^* in [U]^{aleph_1})
(f restriction U^* is continuous).

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