## Some applications of tiny sequences.(English)Zbl 0549.54003

This paper is very interesting. Let X be a topological space. A sequence $$\{P_ n\} (n=1,2,...)$$ of open set families in X is called a tiny sequence in X, if (1) $$\cup P_ n$$ is dense in X for every n, (2) if $$F_ n$$ is a finite subfamily of $$P_ n$$ for each n, then $$\cup_{n}\{\cup F_ n\}$$ is not dense in X. If the cellularity of X is uncountable, then there is a tiny sequence in X. On the other hand, if the cellularity is countable, the $$\pi$$ -weight of X is less than $$2^{\omega}$$ and MA holds, then there are no tiny sequences in X. There is no space of countable $$\pi$$ -weight with a tiny sequence, but there is a regular countable space with a tiny sequence. There is no dyadic space with a tiny sequence, and then any compactification of Suslin line has no tiny sequence. Using the notion of a tiny sequence, the author proves that, under MA, there is a separable closed subspace in $$\omega^*$$ which is not a retract of $$\beta$$ ($$\omega)$$.
Reviewer: K.Iséki

### MSC:

 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54G05 Extremally disconnected spaces, $$F$$-spaces, etc.