Aggregate properties of mappings that are associated with the closedness of
a graph

V. V. Nesterenko

doi:10.15330/ms.43.1.27-35

We study aggregate properties of mappings of two variables that are closed graph relative one of variables. In particular we prove that if $X$ is a topological space, a space $Y$ is a first (second) countable space, $Z$ is a locally compact Hausdorff second countable space, a mapping $f\colon X \times Y \to Z$ such that $f^x$ is continuous for every $x$ with some residual subsets of $X$ and $f_y$ has closed graph for each $y\in Y$, then $C_y(f)=\{x\in X\colon (x, y)\in C(f)\}$ is residual in $X$ for each $y \in Y$ ($C_Y(f)=\{x\in X\colon \{x\}\times Y\subseteq C(f)\}$ is residual in $X$).

1. Kostyrko P. A note on the functions with closed graphs, Cas. Pest. Mat., 94 (1969), №2, 202–205.

2. Baggs I. Functions with a closed graph, Proc. Am. Math. Soc., 43 (1974), 439–442.

3. Alas O.T. A note on functions with a closed graph, Port. Math., 42 (1983), 351–354.

4. Berner A.J. Almost continuous functions with closed graphs, Can. Math. Bull., 25 (1982), 428–434.

5. Dobo.s J. On the set of points of discontinuity for functions with closed graphs, Cas. Pest. Mat., 110 (1985), 60–68.

6. Grande Z. On functions of two variables whose vertical sections have closed graphs, Real Anal. Exch., 27 (2002), 661–668.

7. Marcus S. Sur les fonctions quasicontinues an sens de S. Kempisty, Col-loq. Math., 8 (1961), 47–53.

8. Maslyuchenko V.K., Nesterenko V.V. Joint continuity and quasicontinuity of horizontally quasicontinuous mappings, Ukr. Math. J., 52 (2000), №12, 1711–1714.

9. Maslyuchenko V.K., Mykhajlyuk V.V., Nesterenko, V.V. Symmetrical quasicontinuity of joint quasicontinuous functions, Mat. Stud. 11 (1999), №2, 204–208. (in Ukrainian)

10. Thielman H.P. Types of functions, Amer. Math. Monthly., 60, (1953), 156–161.

11. Nesterenko V.V. Sufficient conditions for the existence of points of symmetrically quasi-continuity and of symmetrically cliquishness of functions of two variables, Carpat. Math. Publ., 3 (2011), №2, 114–119. (in Ukrainian)

12. Fuller R.V. Relations among continuous and various non-continuous functions, Pac. J. Math., 25 (1968), 495–509.

13. Maslyuchenko V.K. On separate and joint modifications of continuity, Mat. Stud. 25 (2006), №2, 213– 218. (in Ukrainian)

14. Calbrix J., Troallic J.P. Applications s.epar.ement continues, C.R. Acad. Sc. Paris. S.er. A., 288 (1979), 647–648.

15. Lipi.nski J.S., Salat T. On the points of quasicontinuity and cliquishness of functions, Czech. Math. J., 21 (1971), №3, 484–489.

16. Dugundji J. Topology, Boston. – Mass: Allyn and Bacon, (1996), 447 p.

17. Engelking R. General topology, Mir, Moskva, (1986), 752 p.

	Aggregate properties of mappings that are associated with the closedness of a graph (in Ukrainian)
Author	V. V. Nesterenko math.analysis.chnu@gmail.com Yuriy Fedkovych Chernivtsi National University
Abstract	We study aggregate properties of mappings of two variables that are closed graph relative one of variables. In particular we prove that if $X$ is a topological space, a space $Y$ is a first (second) countable space, $Z$ is a locally compact Hausdorff second countable space, a mapping $f\colon X \times Y \to Z$ such that $f^x$ is continuous for every $x$ with some residual subsets of $X$ and $f_y$ has closed graph for each $y\in Y$, then $C_y(f)=\{x\in X\colon (x, y)\in C(f)\}$ is residual in $X$ for each $y \in Y$ ($C_Y(f)=\{x\in X\colon \{x\}\times Y\subseteq C(f)\}$ is residual in $X$).
Keywords	continuity; quasi-continuity; cliquishness; horizontally quasi-continuity; transitional function; closed graph
DOI	doi:10.15330/ms.43.1.27-35
Reference	1. Kostyrko P. A note on the functions with closed graphs, Cas. Pest. Mat., 94 (1969), №2, 202–205. 2. Baggs I. Functions with a closed graph, Proc. Am. Math. Soc., 43 (1974), 439–442. 3. Alas O.T. A note on functions with a closed graph, Port. Math., 42 (1983), 351–354. 4. Berner A.J. Almost continuous functions with closed graphs, Can. Math. Bull., 25 (1982), 428–434. 5. Dobo.s J. On the set of points of discontinuity for functions with closed graphs, Cas. Pest. Mat., 110 (1985), 60–68. 6. Grande Z. On functions of two variables whose vertical sections have closed graphs, Real Anal. Exch., 27 (2002), 661–668. 7. Marcus S. Sur les fonctions quasicontinues an sens de S. Kempisty, Col-loq. Math., 8 (1961), 47–53. 8. Maslyuchenko V.K., Nesterenko V.V. Joint continuity and quasicontinuity of horizontally quasicontinuous mappings, Ukr. Math. J., 52 (2000), №12, 1711–1714. 9. Maslyuchenko V.K., Mykhajlyuk V.V., Nesterenko, V.V. Symmetrical quasicontinuity of joint quasicontinuous functions, Mat. Stud. 11 (1999), №2, 204–208. (in Ukrainian) 10. Thielman H.P. Types of functions, Amer. Math. Monthly., 60, (1953), 156–161. 11. Nesterenko V.V. Sufficient conditions for the existence of points of symmetrically quasi-continuity and of symmetrically cliquishness of functions of two variables, Carpat. Math. Publ., 3 (2011), №2, 114–119. (in Ukrainian) 12. Fuller R.V. Relations among continuous and various non-continuous functions, Pac. J. Math., 25 (1968), 495–509. 13. Maslyuchenko V.K. On separate and joint modifications of continuity, Mat. Stud. 25 (2006), №2, 213– 218. (in Ukrainian) 14. Calbrix J., Troallic J.P. Applications s.epar.ement continues, C.R. Acad. Sc. Paris. S.er. A., 288 (1979), 647–648. 15. Lipi.nski J.S., Salat T. On the points of quasicontinuity and cliquishness of functions, Czech. Math. J., 21 (1971), №3, 484–489. 16. Dugundji J. Topology, Boston. – Mass: Allyn and Bacon, (1996), 447 p. 17. Engelking R. General topology, Mir, Moskva, (1986), 752 p.
Pages	27-35
Volume	43
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Aggregate properties of mappings that are associated with the closedness of a graph (in Ukrainian)