## 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

Zbl 0637.03049
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