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On Additive Almost Continuous Functions Under Cpaprismgame, Krzysztof Ciesielski
On Additive Almost Continuous Functions Under Cpaprismgame, Krzysztof Ciesielski
Faculty & Staff Scholarship
We prove that the Covering Property Axiom CPAprismgame, which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function f from R to R whose graph is of measure zero. We also show that, under CPAprismgame, there exists a Hamel basis H for which the set E+(H), of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin's axiom, while it is unknown whether either of them can be constructed …
Separately Nowhere Constant Functions; N-Cube And \Alpha-Prism Densities, Krzysztof Ciesielski
Separately Nowhere Constant Functions; N-Cube And \Alpha-Prism Densities, Krzysztof Ciesielski
Faculty & Staff Scholarship
A function f from a countable product X of of Polish spaces Xi into a Polish space is separately nowhere constant provided it is nowhere constant on every section of X. We show that every continuous separately nowhere constant function is one-to-one on a product of perfect subsets of Xi's. This result is used to distinguish between n-cube density notions for different n\leq\omega, where \omega-cube density is a basic notion behind the Covering Property Axiom CPA formulated by Ciesielski and Pawlikowski. We will also distinguish, for different values of \alpha<\omega1, between the notions of \alpha-prism …