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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