Sequence selection principles for quasi-normal convergence

In Bukovský et al. (2001) [5], the authors proved Theorem 5.10 saying that eleven seemingly different properties of a perfectly normal space X are equivalent. One of the properties says that every Borel image of X into is eventually bounded. B. Tsaban and L. Zdomskyy (in press) [17], have proved that any perfectly normal topological QN-space (for the definition see Bukovský et al. (1991) [4], ) possesses this property, therefore all properties of the theorem. In this paper we simply prove that every perfectly normal topological QN-space possesses another property of that theorem – see Theorems 1 and 3. The main tools of our proof are sequence selection principles for quasi-normal convergence introduced in the paper. Some of introduced principles are worth studying in their own right and we initiate their study. Moreover, one of our main results immediately implies Recławʼs Theorem (Recław, 1997) [14].