The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
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Gertruda Ivanova
ABSTRACT
In this paper, the families of functions continuous with respect to some family of subsets of the real line as a domain are considered. There is proved, among others, that if the family of subsets has some special property, then each function continuous with respect to this family is quasi-continuous (Corollary 2.2.1). We say that two families of subsets of the real line are equivalent if the families of functions which are continuous in generalized sense with respect to these families are equal. Limited considerations to the families which are generalized topologies with additional properties, we find in equivalence classes the smallest and the largest family with respect to inclusion (see Theorem 3.5).
Acknowledgement
The authors are thankful to Professor E. Wagner-Bojakowska for her suggestions during the preparation of the paper.
REFERENCES
[1] Császár, Á.: Generalized topology, generalized continuity, Acta Math. Hungar. 96 (2002), 351–357.10.1023/A:1019713018007Search in Google Scholar
[2] Hashimoto, H.: On the *-topology and its application, Fund. Math. 91 (1976), 5–10.10.4064/fm-91-1-5-10Search in Google Scholar
[3] Ivanova, G.—Karasińska, A.—Wagner-Bojakowska, E.: Families of Darboux functions and topologies having (𝒥*)-property, Topology Appl. 258 (2019), 534–542.10.1016/j.topol.2019.03.021Search in Google Scholar
[4] Ivanova, G.—Wagner-Bojakowska, E.: On some generalization of the notion of continuity. Category case, Lith. Math. J. 59 (2019), 357–365.10.1007/s10986-019-09447-8Search in Google Scholar
[5] Kempisty, S.: Sur les fonctions quasicontinues, Fund. Math. 19 (1932), 184–197.10.4064/fm-19-1-184-197Search in Google Scholar
[6] Kuratowski, K.: Topology, Vol. I., Academic Press, 1966.Search in Google Scholar
[7] Kuratowski, K.—Mostowski, A.: Set theory with an introduction to descriptive set theory, PWN Warszawa, 1976.Search in Google Scholar
[8] Levine, N.: Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36–41.10.1080/00029890.1963.11990039Search in Google Scholar
[9] Martin, N.F.: Generalized condensation points, Duke Math. J. 28 (4) (1961), 507–514.10.1215/S0012-7094-61-02848-4Search in Google Scholar
[10] Neubrunnová, A.: On certain generalizations of the notion of continuity, Matematický Časopis 23 (4) (1973), 374–380.Search in Google Scholar
[11] Oxtoby, J. C.: Measure and Category, Springer-Verlag, New York, 1971.10.1007/978-1-4615-9964-7Search in Google Scholar
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Articles in the same Issue
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
