Affiliation:
1. Ivan Franko National University of Lviv , Ukraine ; Jan Kochanowski University in Kielce , Poland
Abstract
Abstract
A function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn
}
n
∈
ω
of X such that for every n ∈ ω the restriction f ↾ Xn
is continuous. By 𝔠
σ
(resp. 𝔠¯σ)we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality |X| <κ is σ-continuous (resp. ¯σ-continuous). It is clear that ω
1 ≤ 𝔠¯
σ
≤ 𝔠
σ
≤ 𝔠.We prove that 𝔭 ≤ 𝔮0 = 𝔠¯
σ
=min{𝔠
σ
, 𝔟, 𝔮 }≤ 𝔠
σ
≤ min{non(ℳ), non(𝒩)}.
Reference21 articles.
1. [1] ADYAN, S. I.—NOVIKOV, P. S.: On a semicontinuous function, Moskov. Gos. Ped. Inst. Uč. Zap. 138 (1958), 3–10. (In Russian)
2. [2] BANAKH, T.—MACHURA, M.—ZDOMSKYY, L.: On critical cardinalities related to Q-sets, Math. Bull. Shevchenko Sci. Soc. 11 (2014), 21–32.
3. [3] BLASS, A.: Combinatorial cardinal characteristics of the continuum. In: Handbook of Set Theory, Vols. 1,2,3, Springer, Dordrecht, 2010, pp. 395–489.10.1007/978-1-4020-5764-9_7
4. [4] CICHOŃ, J.—MORAYNE, M.—PAWLIKOWSKI, J.—SOLECKI, S.: Decomposing Baire functions, J. Symbolic Logic 56 (1991), 1273–1283.10.2307/2275474
5. [5] DARJI, U.: Countable decomposition of derivatives and Baire 1 functions, J. Appl. Anal. 2 (1996), 119–124.10.1515/JAA.1996.119