Title: Graph Quasicontinuous Functions and Densely Continuous Forms
Author(s): Lubica Hola, Dusan Holy
Pages: 155-161
Cite as:
Lubica Hola, Dusan Holy, Graph Quasicontinuous Functions and Densely Continuous Forms, Int. J. Anal. Appl., 14 (2) (2017), 155-161.

Abstract


Let $X, Y$ be topological spaces. A function $f: X \to Y$ is said to be graph quasicontinuous if there is a quasicontinuous function $g: X \to Y$ with the graph of $g$ contained in the closure of the graph of $f$. There is a close relation between the notions of graph quasicontinuous functions and minimal usco maps as well as the notions of graph quasicontinuous functions and densely continuous forms. Every function with values in a compact Hausdorff space is graph quasicontinuous; more generally every locally compact function is graph quasicontinuous.

Full Text: PDF

 

References


  1. G. Beer: Topologies on closed and closed convex sets, Kluwer Academic Publisher 1993.

  2. C. Berge: Topological Spaces, Oliver and Boyd, Edinburgh 1963.

  3. J. Bors´ık: Points of continuity, quasicontinuity and cliquishness, Rend. Ist. Math. Univ. Trieste 26 (1994), 5–20.

  4. A. Bouziad: Every Cech-analytic Baire semitopological group is a topological group, Proc. Amer. Math. Soc. 124 (1996), 953–959.

  5. L. Drewnowski and I. Labuda: On minimal upper semicontinuous compact valued maps, Rocky Mountain J. Math. 20 (1990), 737–752.

  6. A. Crannell, M. Frantz and M. LeMasurier: Closed relations and equivalence classes of quasicontinuous functions, Real Anal. Exch. 31 (2006/2007), 409-424.

  7. J.P.R. Christensen: Theorems of Namioka and R.E. Johnson type for upper semicontinuous and compact valued mappings, Proc. Amer. Math. Soc. 86 (1982), 649–655.

  8. R. Engelking: General Topology, PWN 1977.

  9. R.V. Fuller: Set of points of discontinuity, Proc. Amer. Math. Soc. 38 (1973), 193–197.

  10. J.R. Giles and M.O. Bartlett, Modified continuity and a generalization of Michael’s selection theorem, Set-Valued Anal. 1 (1993), 247-268.

  11. Z. Grande: A note on the graph quasicontinuity, Demonstr. Math. 39 (2006), 515–518.

  12. L. Holá and D. Hol´ y: Minimal usco maps, densely continuous forms and upper semicontinuous functions, Rocky Mountain J. Math. 39 (2009), 545–562.

  13. L. Holá and D. Hol´ y: New characterization of minimal CUSCO maps, Rocky Mount. Math. J. 44 (2014), 1851–1866.

  14. S.T. Hammer, R.A. McCoy: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266.

  15. L. Holá and B. Novotn´ y: Subcontinuity of multifunctions, Math. Slovaca 62 (2012), 345-362.

  16. S. Kempisty: Sur les fonctions quasi-continues, Fund. Math. 19 (1932), 184–197.

  17. A. Lechicki and S. Levi: Extensions of semicontinuous multifunctions, Forum Math. 2 (1990), 341–360.

  18. M. Matejdes, Minimality of multifunctions, Real Anal. Exch. 32 (2007), 519–526.

  19. A. Mikucka: Graph quasi-continuity, Demonstr. Math. 36 (2003), 483–494.

  20. W.B. Moors: Any semitopological group that is homeomorphic to a product of Cech-complete spaces is a topological group, Set-Valued Var. Anal. 21 (2013), 627-633.

  21. W.B. Moors: Semitopological groups, Bouziad spaces and topological groups, Topology Appl. 160 (2013), 2038-2048.

  22. T. Neubrunn: Quasi-continuity, Real Anal. Exch. 14 (1988), 259–306.

  23. H. Vaughan: On locally compact metrizable spaces, Bull. Amer. Math. Soc. 43 (1937), 532–535.