Selection principle S1 and combinatorics of open covers

Selectors for dense subsets of function spaces

“…In [3] similar results are presented for S f in . Consequently, one can easily see that plenty of our results may be proved also for S f in .…”

“…In [3] the author (Theorems 6.2 and 6.5 part 5)) 2 has proved Theorem 3.1 (L. Bukovský). Assume that Φ is one of the symbols Ω and Γ, and Ψ is one of the symbols O, Ω, Γ.…”

“…A sequence of real functions f n : n ∈ ω converges to a real function f in this topology if it converges pointwise, i.e., if lim n→∞ f n (x) = f (x) for each x ∈ X. Similarly as in [3] we introduce the following properties of a family F of real functions and a function h ∈ X R:…”

“…That was M. Sakai [19] who immediately exploited this idea. The starting point of our study presented in this paper were the results of [3] and the investigation started in [12].…”

Ideal approach to convergence in functional spaces

“…By part (1), we have Fin β M ≤ K Fin α M , and by part (5), it is enough to show that Fin α ≤ K Fin β . However, this follows by results by M. Katětov [26]: 9 He has shown in his Theorem 7.1 that considering all pointwise limits of all continuous functions on a topological space with respect to Fin ξ , one obtains exactly ξ-th Baire class of functions. If Fin α ≤ K Fin β , then by M. Katětov [26], Proposition 4.2, the α-th Baire class of functions is below β-th Baire class of functions.…”