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


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