A characterization of the uniform convergence points set of some convergent sequence of functions
-
Olena Karlova
Abstract
We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X, then A is the set of points of the uniform convergence for some convergent sequence (fn)n∈ω of functions fn : X → ℝ if and only if A is Gδ-set which contains all isolated points of X. This result generalizes a theorem of Ján Borsík published in 2019.
Acknowledgement
I am very grateful to the reviewers for their careful reading of the manuscript and valuable remarks which allowed to improve the paper.
Dedicated to the memory of Ján Borsík
(Communicated by Tomasz Natkaniec )
References
[1] Borsík, J.: Points of uniform convergence and quasicontinuity, Eur. J. Math. 5 (2019), 174–185.10.1007/s40879-018-0303-4Suche in Google Scholar
[2] Comfort, W. W.—Garcia-Ferreira, S.: Resolvability: a selective survey and some new results, Topology Appl. 74 (1996), 149–167.10.1016/S0166-8641(96)00052-1Suche in Google Scholar
[3] van Douwen, E. K.: Applications of maximal topologies, Topology Appl. 51(2) (1993), 125–139.10.1016/0166-8641(93)90145-4Suche in Google Scholar
[4] Drahovský, Š.—Šalát, T.—Toma, V.: Points of uniform convergence and oscillation of sequences of functions, Real Anal. Exchange 20 (1994–95), 753–767.10.2307/44152557Suche in Google Scholar
[5] Engelking, R.: General Topology, Revised and completed edition, Heldermann Verlag, Berlin, 1989.Suche in Google Scholar
[6] Hahn, H.: Über die Menge der Konvergenzpunkte einer Funktionenfolge, Arch. d. Math. u. Phys. 28 (1919–1920), 34–45.10.1007/978-3-7091-6590-4_28Suche in Google Scholar
[7] Holá, L.—Holý, D.: Pointwise convergence of quasicontinuous functions and Baire spaces, Rocky Mountain J. Math. 41 (2011), 1883–1894.10.1216/RMJ-2011-41-6-1883Suche in Google Scholar
[8] Maslyuchenko, V. K.—Mykhaylyuk, V. V.—Sobchuk, O. V.: Inverse problems of the theory of separately continuous mappings, Translation from Ukrainian J. Math. 44(9) (1992), 1209–1220.10.1007/BF01058371Suche in Google Scholar
[9] Natkaniec, T.—Wesołowska, J.: On the convergence ofω1sequences of real functions, Acta Math. Hungar. 90 (2001), 335–350.10.1023/A:1010691315699Suche in Google Scholar
[10] Natkaniec, T.—Wesołowska, J.: Sets of ideal convergence of sequences of quasicontinuous functions, J. Math. Anal. Appl. 423 (2015), 924–939.10.1016/j.jmaa.2014.10.029Suche in Google Scholar
[11] Sierpiński, W.: Sur ľensemble des points de convergence ďune suite de fonctions continues, Fund. Math. 2 (1921), 41–49.10.4064/fm-2-1-41-49Suche in Google Scholar
[12] Wesołowska, J.: On set of convergence of convergence points of transfnite sequence of quasi-continuous functions, Tatra Mt. Math. Publ. 28 (2004), 21–27.Suche in Google Scholar
[13] Wesołowska, J.: On sets of convergence points of sequences of some real functions, Real Anal. Exchange 25 (1999–2000), 937–942.10.2307/44154048Suche in Google Scholar
[14] Wesołowska, J.: On sets of discrete convergence points of sequences of real functions, Real Anal. Exchange 29 (2003–2004), 107–120.10.14321/realanalexch.29.1.0107Suche in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Prof. RNDr. Michal Fečkan, DrSc. – Sexagenarian?
- Tribonacci numbers with two blocks of repdigits
- Padovan numbers that are concatenations of two distinct repdigits
- On the 2-rank and 4-rank of the class group of some real pure quartic number fields
- A general inverse matrix series relation and associated polynomials – II
- Some hardy type inequalities with finsler norms
- Starlikeness and convexity of the product of certain multivalent functions with higher-order derivatives
- Block Hessenberg matrices and spectral transformations for matrix orthogonal polynomials on the unit circle
- How is the period of a simple pendulum growing with increasing amplitude?
- Fourier transforms of convolution operators on orlicz spaces
- Some characterizations of property of trans-Sasakian 3-manifolds
- P-Adic metric preserving functions and their analogues
- On statistical convergence of sequences of closed sets in metric spaces
- A characterization of the uniform convergence points set of some convergent sequence of functions
- A nonparametric estimation of the conditional ageing intensity function in censored data: A local linear approach
- Donsker’s fuzzy invariance principle under the Lindeberg condition
- Characterization of generalized Gamma-Lindley distribution using truncated moments of order statistics
- Matrix variate pareto distributions
- Global exponential periodicity and stability of neural network models with generalized piecewise constant delay
- Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds
Artikel in diesem Heft
- Regular papers
- Prof. RNDr. Michal Fečkan, DrSc. – Sexagenarian?
- Tribonacci numbers with two blocks of repdigits
- Padovan numbers that are concatenations of two distinct repdigits
- On the 2-rank and 4-rank of the class group of some real pure quartic number fields
- A general inverse matrix series relation and associated polynomials – II
- Some hardy type inequalities with finsler norms
- Starlikeness and convexity of the product of certain multivalent functions with higher-order derivatives
- Block Hessenberg matrices and spectral transformations for matrix orthogonal polynomials on the unit circle
- How is the period of a simple pendulum growing with increasing amplitude?
- Fourier transforms of convolution operators on orlicz spaces
- Some characterizations of property of trans-Sasakian 3-manifolds
- P-Adic metric preserving functions and their analogues
- On statistical convergence of sequences of closed sets in metric spaces
- A characterization of the uniform convergence points set of some convergent sequence of functions
- A nonparametric estimation of the conditional ageing intensity function in censored data: A local linear approach
- Donsker’s fuzzy invariance principle under the Lindeberg condition
- Characterization of generalized Gamma-Lindley distribution using truncated moments of order statistics
- Matrix variate pareto distributions
- Global exponential periodicity and stability of neural network models with generalized piecewise constant delay
- Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds
