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A tree \(\pi \)-base for \(\mathbb R^{\ast }\) without cofinal branches. (English) Zbl 1121.54057
Summary: We prove an analogue to P. L. Dordal’s result in [J. Symb. Log. 52, 651–664 (1987; Zbl 0637.03049)]. He obtained a model of ZFC in which there is a tree \(\pi \)-base for \(\mathbb N^{\ast }\) with no \(\omega _{2}\) branches yet of height \(\omega _{2}\). We establish that this is also possible for \(\mathbb R^{\ast }\) using a natural modification of Mathias forcing.
MSC:
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54A35 Consistency and independence results in general topology
03E17 Cardinal characteristics of the continuum
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