We prove an analogue to Dordal’s result in P.L. Dordal, , J. Symbolic Logic (1980), 651–664. He obtained a model of ZFC in which there is a tree -base for with no branches yet of height . We establish that this is also possible for using a natural modification of Mathias forcing.
We prove that, assuming MA, every crowded space is -resolvable if it satisfies one of the following properties: (1) it contains a -network of cardinality constituted by infinite sets, (2) , (3) is a Baire space and and (4) is a Baire space and has a network with cardinality and such that the collection of the finite elements in it constitutes a -locally finite family. Furthermore, we prove that the existence of a Baire irresolvable space is equivalent to the existence of...
We study the problem in the title and show that it is equivalent to the fact that every set of reals is an increasing union of measurable sets. We also show the relationship of it with Sierpi'nski sets.
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