Composition and discrete convergence

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1 Composition and discrete convergence Jaroslav Šupina joint work with Dávid Uhrik Institute of Mathematics Faculty of Science P.J. Šafárik University in Košice 4 th of July 2018

2 Šupina J., On Ohta Sakai s properties of a topological space, Topology Appl. 190 (2015), Let F, G be families of real-valued functions on a set X. We say that X has a property DL(F, G) if any function from F is a discrete limit of a sequence of functions from G.

3 Šupina J., On Ohta Sakai s properties of a topological space, Topology Appl. 190 (2015), Let F, G be families of real-valued functions on a set X. We say that X has a property DL(F, G) if any function from F is a discrete limit of a sequence of functions from G.

4 Convergence of f n : n ω, f n, f : X R Pointwise convergence f n f ( x X)( ε > 0)( n 0 )( n ω)(n n 0 f n(x) f(x) < ε) Monotone convergence f n f f n f f n f f n f ( n ω) f n f n+1 f n f f n f ( n ω) f n f n+1 Quasi-normal (equal) convergence QN f n QN f there exists ε n : n ω converging to 0 such that ( x X)( n 0 )( n ω)(n n 0 f n(x) f(x) < ε n) Discrete convergence D f n D f ( x X)( n 0 )( n ω)(n n 0 f n(x) = f(x))

5 Šupina J., On Ohta Sakai s properties of a topological space, Topology Appl. 190 (2015), Let F, G be families of real-valued functions on a set X. We say that X has a property DL(F, G) if any function from F is a discrete limit of a sequence of functions from G.

6 X R the family of all real-valued functions on X X [0, 1] the family of all functions on X with values in [0, 1] B B 1 M 0 2 U L C(X) the family of all Borel functions on X the family of all first Baire class functions on X the family of all 0 2-measurable functions on X the family of all upper semicontinuous functions on X the family of all lower semicontinuous functions on X the family of all continuous functions on X F X R F = F X [0, 1]

7 Bukovská Z., Quasinormal convergence, Math. Slovaca 41 (1991), Császár Á. and Laczkovich M., Some remarks on discrete Baire classes, Acta Math. Acad. Sci. Hungar. 33 (1979), Theorem Let X be a normal space, f : X R. The following are equivalent. (1) f is a discrete limit of a sequence of continuous functions on X. (2) f is a quasi-normal limit of a sequence of continuous functions on X. (3) There is a sequence F n : n ω of closed subsets of X such that f F n is continuous on F n for any n ω and X = n ω Fn.

8 Jayne J.E. and Rogers C.A., First level Borel functions and isomorphisms, J. Math. Pures et Appl. 61 (1982), Theorem If A is an analytic subset of a Polish space then A has DL(M 0 2, C(X)). Bukovský L., Recław I. and Repický M., Spaces not distinguishing convergences of real valued functions, Topology Appl. 112 (2001), Tsaban B. and Zdomskyy L., Hereditary Hurewicz spaces and Arhangel skiǐ sheaf amalgamations, J. Eur. Math. Soc. (JEMS), 14 (2012), Proposition Any perfectly normal QN-space has DL(M 0 2, C(X)). Theorem A perfectly normal space X is a QN-space if and only if X has Hurewicz property and DL(B 1, C(X)).

9 Jayne J.E. and Rogers C.A., First level Borel functions and isomorphisms, J. Math. Pures et Appl. 61 (1982), Theorem If A is an analytic subset of a Polish space then A has DL(M 0 2, C(X)). Bukovský L., Recław I. and Repický M., Spaces not distinguishing convergences of real valued functions, Topology Appl. 112 (2001), Tsaban B. and Zdomskyy L., Hereditary Hurewicz spaces and Arhangel skiǐ sheaf amalgamations, J. Eur. Math. Soc. (JEMS), 14 (2012), Proposition Any perfectly normal QN-space has DL(M 0 2, C(X)). Theorem A perfectly normal space X is a QN-space if and only if X has Hurewicz property and DL(B 1, C(X)).

10 Jayne J.E. and Rogers C.A., First level Borel functions and isomorphisms, J. Math. Pures et Appl. 61 (1982), Theorem If A is an analytic subset of a Polish space then A has DL(M 0 2, C(X)). Bukovský L., Recław I. and Repický M., Spaces not distinguishing convergences of real valued functions, Topology Appl. 112 (2001), Tsaban B. and Zdomskyy L., Hereditary Hurewicz spaces and Arhangel skiǐ sheaf amalgamations, J. Eur. Math. Soc. (JEMS), 14 (2012), Proposition Any perfectly normal QN-space has DL(M 0 2, C(X)). Theorem A perfectly normal space X is a QN-space if and only if X has Hurewicz property and DL(B 1, C(X)).

11 Convergence of f n : n ω, f n, f : X R Pointwise convergence f n f ( x X)( ε > 0)( n 0 )( n ω)(n n 0 f n(x) f(x) < ε) Monotone convergence f n f f n f f n f f n f ( n ω) f n f n+1 f n f f n f ( n ω) f n f n+1 Quasi-normal (equal) convergence QN f n QN f there exists ε n : n ω converging to 0 such that ( x X)( n 0 )( n ω)(n n 0 f n(x) f(x) < ε n) Discrete convergence D f n D f ( x X)( n 0 )( n ω)(n n 0 f n(x) = f(x))

12 Bukovský L., Recław I. and Repický M., Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology Appl. 41 (1991), A topological space X is a QN-space if each sequence of continuous real-valued functions converging to zero on X is converging quasi-normally. Tychonoff QN-space is zero-dimensional any QN-subset of a metric separable space is perfectly meager perfectly normal QN-space has Hurewicz property non(qn-space) = b b-sierpiński set is a QN-space (exists under b = cov(n ) = cof(n )) Recław I., Metric spaces not distinguishing pointwise and quasinormal convergence of real functions, Bull. Acad. Polon. Sci. 45 (1997), Miller A.W., On the length of Borel hierarchies, Ann. Math. Logic 16 (1979), perfectly normal QN-space is a σ-set the theory ZFC + any QN-space is countable is consistent

13 Bukovský L., Recław I. and Repický M., Spaces not distinguishing convergences of real valued functions, Topology Appl. 112 (2001), Proposition Let X be a perfectly normal space. The following are equivalent. (1) X is a σ-set with DL(M 0 2, C(X)). (2) X possesses DL(B 1, C(X)). (3) X possesses DL(B, C(X)).

14 Cichoń J. and Morayne M., Universal functions and generalized classes of functions, Proc. Amer. Math. Soc. 102 (1988), Cichoń J., Morayne M., Pawlikowski J. and Solecki S. Decomposing Baire functions, J. Symbolic Logic 56 (1991), Let F, G be families of real-valued functions on a set X. dec(f, G) denotes the minimal cardinal κ such that any function from F can be decomposed into κ many functions from G. X is usually a Polish space.

15 Cichoń J. and Morayne M., Universal functions and generalized classes of functions, Proc. Amer. Math. Soc. 102 (1988), Cichoń J., Morayne M., Pawlikowski J. and Solecki S. Decomposing Baire functions, J. Symbolic Logic 56 (1991), Let F, G be families of real-valued functions on a set X. dec(f, G) denotes the minimal cardinal κ such that any function from F can be decomposed into κ many functions from G. X is usually a Polish space.

16 Cichoń J. and Morayne M., Universal functions and generalized classes of functions, Proc. Amer. Math. Soc. 102 (1988), Cichoń J., Morayne M., Pawlikowski J. and Solecki S. Decomposing Baire functions, J. Symbolic Logic 56 (1991), Let F, G be families of real-valued functions on a set X. dec(f, G) denotes the minimal cardinal κ such that any function from F can be decomposed into κ many functions from G. X is usually a Polish space.

17 Theorem (a) Let X be a topological space. Then DL(U, C(X)) DL(L, C(X)) DL(Ũ, C(X)) DL( L, C(X)) ( Y X) DL(U, C(Y )) ( Y X) DL(L, C(Y )). (b) Let X be a perfectly normal space. Then DL(U, L) DL(L, U) DL(L, C(X)) DL(B 1, C(X)) DL(B, C(X)).

18 Theorem (a) Let X be a topological space. Then DL(U, C(X)) DL(L, C(X)) DL(Ũ, C(X)) DL( L, C(X)) ( Y X) DL(U, C(Y )) ( Y X) DL(L, C(Y )). (b) Let X be a perfectly normal space. Then DL(U, L) DL(L, U) DL(L, C(X)) DL(B 1, C(X)) DL(B, C(X)).

19 F σ-measurable function f : X [0, 1] Lindenbaum s Theorem: there are lower semicontinuous functions g : [0, 1] [0, 1], h : X [0, 1] such that f = g h for h: F n : n ω of closed subsets of X, h F n is continuous on F n, X = F n n ω f F n = g h F n is lower semicontinuous on F n for g: F n,m : m ω of closed subsets of X, f F n,m = g h F n,m is continuous on F n,m, F n = F n,m m ω

20 F σ-measurable function f : X [0, 1] Lindenbaum s Theorem: there are lower semicontinuous functions g : [0, 1] [0, 1], h : X [0, 1] such that f = g h for h: F n : n ω of closed subsets of X, h F n is continuous on F n, X = F n n ω f F n = g h F n is lower semicontinuous on F n for g: F n,m : m ω of closed subsets of X, f F n,m = g h F n,m is continuous on F n,m, F n = F n,m m ω

21 F σ-measurable function f : X [0, 1] Lindenbaum s Theorem: there are lower semicontinuous functions g : [0, 1] [0, 1], h : X [0, 1] such that f = g h for h: F n : n ω of closed subsets of X, h F n is continuous on F n, X = F n n ω f F n = g h F n is lower semicontinuous on F n for g: F n,m : m ω of closed subsets of X, f F n,m = g h F n,m is continuous on F n,m, F n = F n,m m ω

22 F σ-measurable function f : X [0, 1] Lindenbaum s Theorem: there are lower semicontinuous functions g : [0, 1] [0, 1], h : X [0, 1] such that f = g h for h: F n : n ω of closed subsets of X, h F n is continuous on F n, X = F n n ω f F n = g h F n is lower semicontinuous on F n for g: F n,m : m ω of closed subsets of X, f F n,m = g h F n,m is continuous on F n,m, F n = F n,m m ω

23 F σ-measurable function f : X [0, 1] Lindenbaum s Theorem: there are lower semicontinuous functions g : [0, 1] [0, 1], h : X [0, 1] such that f = g h for h: F n : n ω of closed subsets of X, h F n is continuous on F n, X = F n n ω f F n = g h F n is lower semicontinuous on F n for g: F n,m : m ω of closed subsets of X, f F n,m = g h F n,m is continuous on F n,m, F n = F n,m m ω

24 Lindenbaum A., Sur les superpositions des fonctions représentables analytiquement, Fund. Math. 23 (1934), Cichoń J., Morayne M., Pawlikowski J. and Solecki S. Decomposing Baire functions, J. Symbolic Logic 56 (1991),

25 Convergence of f n : n ω, f n, f : X R Pointwise convergence f n f ( x X)( ε > 0)( n 0 )( n ω)(n n 0 f n(x) f(x) < ε) Monotone convergence f n f f n f f n f f n f ( n ω) f n f n+1 f n f f n f ( n ω) f n f n+1 Quasi-normal (equal) convergence QN f n QN f there exists ε n : n ω converging to 0 such that ( x X)( n 0 )( n ω)(n n 0 f n(x) f(x) < ε n) Discrete convergence D f n D f ( x X)( n 0 )( n ω)(n n 0 f n(x) = f(x))

26 Baire 1899: B 0(X) = C(X) B α(x) = { f : f n f f n β<α } B β (X) Young 1910: L 0(X) = U 0(X) = C(X) L α(x) = { f : f n f f n β<α } U β (X) β<α { U α(x) = f : f n f f n } L β (X) L 1 L 2 L α C(X) B 1 B 2 B α U 1 U 2 U α

27 MΣ 0 α(x) = { f : ( U open in [0, 1]) f 1 (U) Σ 0 α(x) } MΣ 0 α (X) = { f : ( r [0, 1]) f 1 ((r, 1]) Σ 0 α (X)} MΣ 0 α(x) = { f : ( r [0, 1]) f 1 ([0, r)) Σ 0 α(x) } B α(x) = MΣ 0 α+1(x) L α(x) = MΣ 0 α (X) U α(x) = MΣ 0 α(x) MΣ 0 1 MΣ 0 2 MΣ 0 α MΣ 0 1 MΣ 0 2 MΣ 0 3 MΣ 0 α MΣ 0 1 MΣ 0 2 MΣ 0 α CSÁSZÁR, Á., LACZKOVICH, M.: Some remarks on discrete Baire classes, Acta Math. Acad. Sci. Hung. 33 (1979), SIKORSKI, R.: Funkcje Rzeczywiste I, Panstwowe Wydawnictwo Naukowe 1958.

28 Uhrik D., Compositions of discontinuous functions, Master Thesis, UPJŠ, Košice, Theorem (A. Lindenbaum) Let X be a perfectly normal topological space, α < ω 1. Then there exists a g L 1 ([0, 1]) such that L α+1 (X) = g U α(x, [0, 1]). Corollary Let X be a perfectly normal topological space, α < ω 1. Then we have L α+1 = L 1 U α = L 1 L α = L 1 B α U α+1 = U 1 L α = U 1 U α = U 1 B α Theorem (A. Lindenbaum) Let X be a perfectly normal topological space, 0 < α, β < ω 1. Then L β+α (X) = L α([0, 1]) L β (X, [0, 1]).

29 Uhrik D., Compositions of discontinuous functions, Master Thesis, UPJŠ, Košice, Theorem (A. Lindenbaum) Let X be a perfectly normal topological space, α < ω 1. Then there exists a g L 1 ([0, 1]) such that L α+1 (X) = g U α(x, [0, 1]). Corollary Let X be a perfectly normal topological space, α < ω 1. Then we have L α+1 = L 1 U α = L 1 L α = L 1 B α U α+1 = U 1 L α = U 1 U α = U 1 B α Theorem (A. Lindenbaum) Let X be a perfectly normal topological space, 0 < α, β < ω 1. Then L β+α (X) = L α([0, 1]) L β (X, [0, 1]).

30 Uhrik D., Compositions of discontinuous functions, Master Thesis, UPJŠ, Košice, Theorem (A. Lindenbaum) Let X be a perfectly normal topological space, α < ω 1. Then there exists a g L 1 ([0, 1]) such that L α+1 (X) = g U α(x, [0, 1]). Corollary Let X be a perfectly normal topological space, α < ω 1. Then we have L α+1 = L 1 U α = L 1 L α = L 1 B α U α+1 = U 1 L α = U 1 U α = U 1 B α Theorem (A. Lindenbaum) Let X be a perfectly normal topological space, 0 < α, β < ω 1. Then L β+α (X) = L α([0, 1]) L β (X, [0, 1]).

31 Cichoń J., Morayne M., Pawlikowski J. and Solecki S., Decomposing Baire functions, J. Symbolic Logic 56 (1991), Set S [0, 1], dense in [0, 1], φ : ω S T [0, 1] is a homeomorphism. Definition Let X be a topological space, (f i ) i<ω be a sequence of functions from X [0, 1]. The coding function of (f i ) i<ω from X to ω [0, 1] is defined by f(x) = (f i (x)) i<ω. Proposition Let X be a perfectly normal topological space, α < ω 1, (f i ) i<ω be a sequence of functions from L α(x, S). Then φ f L α(x). Proposition The function s : ω [0, 1] [0, 1] defined by is in L 1 ( ω [0, 1]). s((t i ) i<ω ) = sup {t i : i < ω}

32 Cichoń J., Morayne M., Pawlikowski J. and Solecki S., Decomposing Baire functions, J. Symbolic Logic 56 (1991), Set S [0, 1], dense in [0, 1], φ : ω S T [0, 1] is a homeomorphism. Definition Let X be a topological space, (f i ) i<ω be a sequence of functions from X [0, 1]. The coding function of (f i ) i<ω from X to ω [0, 1] is defined by f(x) = (f i (x)) i<ω. Proposition Let X be a perfectly normal topological space, α < ω 1, (f i ) i<ω be a sequence of functions from L α(x, S). Then φ f L α(x). Proposition The function s : ω [0, 1] [0, 1] defined by is in L 1 ( ω [0, 1]). s((t i ) i<ω ) = sup {t i : i < ω}

33 Cichoń J., Morayne M., Pawlikowski J. and Solecki S., Decomposing Baire functions, J. Symbolic Logic 56 (1991), Set S [0, 1], dense in [0, 1], φ : ω S T [0, 1] is a homeomorphism. Definition Let X be a topological space, (f i ) i<ω be a sequence of functions from X [0, 1]. The coding function of (f i ) i<ω from X to ω [0, 1] is defined by f(x) = (f i (x)) i<ω. Proposition Let X be a perfectly normal topological space, α < ω 1, (f i ) i<ω be a sequence of functions from L α(x, S). Then φ f L α(x). Proposition The function s : ω [0, 1] [0, 1] defined by is in L 1 ( ω [0, 1]). s((t i ) i<ω ) = sup {t i : i < ω}

34 Cichoń J., Morayne M., Pawlikowski J. and Solecki S., Decomposing Baire functions, J. Symbolic Logic 56 (1991), Set S [0, 1], dense in [0, 1], φ : ω S T [0, 1] is a homeomorphism. Definition Let X be a topological space, (f i ) i<ω be a sequence of functions from X [0, 1]. The coding function of (f i ) i<ω from X to ω [0, 1] is defined by f(x) = (f i (x)) i<ω. Proposition Let X be a perfectly normal topological space, α < ω 1, (f i ) i<ω be a sequence of functions from L α(x, S). Then φ f L α(x). Proposition The function s : ω [0, 1] [0, 1] defined by is in L 1 ( ω [0, 1]). s((t i ) i<ω ) = sup {t i : i < ω}

35 Thanks for Your attention!

Set-theoretic methods in topology and real

birthday of Lev Bukovsky KOSICE ˇ 9.9. - 13.

David Goldstern Martin Hart Klaas Pieter

Sakai Masami Zdomskyy Lyubomyr Organizing

37 Set-theoretic methods in topology and real functions theory, dedicated to 80th birthday of Lev Bukovsky KOSICE ˇ Scientific Committee Chodounský David Goldstern Martin Hart Klaas Pieter Holá Ľubica Jech Thomas Repický Miroslav Sakai Masami Zdomskyy Lyubomyr Organizing Committee Eliáš Peter Repický Miroslav Šupina Jaroslav

Every Lusin set is undetermined in the point-open game

F U N D A M E N T A MATHEMATICAE 144 (1994) Every Lusin set is undetermined in the point-open game by Ireneusz Recław (Gdańsk) Abstract. We show that some classes of small sets are topological versions