Sequence selection principles for quasi-normal convergence
The authors introduce and investigate the following sequence selection principles for quasi-normal and discrete convergence. Let \(A\), \(B\) denote one of the following types of convergence: \(P\)-pointwise; \(Q\)-quasi-normal; \(D\)-discrete. A topological space \(X\) satisfies the sequence selection principle \(ASB\) if for any functions \(f,f_nf^n_m:X\to \mathbb{R}\), \(n,m\in\omega\), with: (a)~\(f_n\) \(A\)-converges to \(f\) on \(X\); (b)~for every \(n\), \(f^n_m\) \(A\)-converges to \(f_n\); (c)~all \(f^n_m\) are continuous, there exists an unbounded \(\beta:\omega\to\omega\) such that \(f^n_{\beta(n)}\) \(B\)-converges to \(f\) on \(X\). Using this notion the authors obtain new and interesting results on QN-spaces. (Recall that a topological space \(X\) is a QN-space if each sequence of continuous real valued functions converging to zero on \(X\) converges to zero quasi-normally as well.) The main results: (1)~any QN-space satisfies the principle \(QSQ\) (Theorem~3); and (2)~if \(X\) is a normal topological space satisfying \(QSQ\), then every \(\Delta^0_2\)-measurable function from \(X\) into \([0,1]\) is a quasi-normal limit of a sequence of continuous functions, therefore also a discrete limit of such a sequence. These results allow to generalize some results of I. Recław, and B.~Tsaban and L.~Zdomskyy.
selection principles for sequences of real functions
scientific article; zbMATH DE number 5986565