Category theorems concerning Z-density continuous functions

K. Ciesielski; L. Larson

Fundamenta Mathematicae (1991)

  • Volume: 140, Issue: 1, page 79-85
  • ISSN: 0016-2736

Abstract

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The ℑ-density topology on ℝ is a refinement of the natural topology. It is a category analogue of the density topology [9, 10]. This paper is concerned with ℑ-density continuous functions, i.e., the real functions that are continuous when the ℑ-densitytopology is used on the domain and the range. It is shown that the family of ordinary continuous functions f: [0,1]→ℝ which have at least one point of ℑ-density continuity is a first category subset of C([0,1])= f: [0,1]→ℝ: f is continuous equipped with the uniform norm. It is also proved that the class of ℑ-density continuous functions, equipped with the topology of uniform convergence, is of first category in itself. These results remain true when the ℑ-density topology is replaced by the deep ℑ-density topology.

How to cite

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Ciesielski, K., and Larson, L.. "Category theorems concerning Z-density continuous functions." Fundamenta Mathematicae 140.1 (1991): 79-85. <http://eudml.org/doc/211931>.

@article{Ciesielski1991,
abstract = {The ℑ-density topology $T_ℑ$ on ℝ is a refinement of the natural topology. It is a category analogue of the density topology [9, 10]. This paper is concerned with ℑ-density continuous functions, i.e., the real functions that are continuous when the ℑ-densitytopology is used on the domain and the range. It is shown that the family $C_ℑ$ of ordinary continuous functions f: [0,1]→ℝ which have at least one point of ℑ-density continuity is a first category subset of C([0,1])= f: [0,1]→ℝ: f is continuous equipped with the uniform norm. It is also proved that the class $C_ℑℑ$ of ℑ-density continuous functions, equipped with the topology of uniform convergence, is of first category in itself. These results remain true when the ℑ-density topology is replaced by the deep ℑ-density topology. },
author = {Ciesielski, K., Larson, L.},
journal = {Fundamenta Mathematicae},
keywords = {ℑ-density topology; ℑ-density continuous functions; first category sets; category analogue; density topology; uniform norm},
language = {eng},
number = {1},
pages = {79-85},
title = {Category theorems concerning Z-density continuous functions},
url = {http://eudml.org/doc/211931},
volume = {140},
year = {1991},
}

TY - JOUR
AU - Ciesielski, K.
AU - Larson, L.
TI - Category theorems concerning Z-density continuous functions
JO - Fundamenta Mathematicae
PY - 1991
VL - 140
IS - 1
SP - 79
EP - 85
AB - The ℑ-density topology $T_ℑ$ on ℝ is a refinement of the natural topology. It is a category analogue of the density topology [9, 10]. This paper is concerned with ℑ-density continuous functions, i.e., the real functions that are continuous when the ℑ-densitytopology is used on the domain and the range. It is shown that the family $C_ℑ$ of ordinary continuous functions f: [0,1]→ℝ which have at least one point of ℑ-density continuity is a first category subset of C([0,1])= f: [0,1]→ℝ: f is continuous equipped with the uniform norm. It is also proved that the class $C_ℑℑ$ of ℑ-density continuous functions, equipped with the topology of uniform convergence, is of first category in itself. These results remain true when the ℑ-density topology is replaced by the deep ℑ-density topology.
LA - eng
KW - ℑ-density topology; ℑ-density continuous functions; first category sets; category analogue; density topology; uniform norm
UR - http://eudml.org/doc/211931
ER -

References

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  1. [1] A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Math. 659, Springer, 1978. Zbl0382.26002
  2. [2] K. Ciesielski and L. Larson, Analytic functions are ℑ-density continuous, submitted. Zbl0826.26011
  3. [3] K. Ciesielski and L. Larson, Baire classification of ℑ-approximately and ℑ-density continuous functions, submitted. Zbl0844.26002
  4. [4] K. Ciesielski and L. Larson, The space of density continuous functions, Acta Math. Hungar., to appear. Zbl0757.26006
  5. [5] K. Ciesielski and L. Larson, Various continuities with the density, ℑ-density and ordinary topologies on ℝ, Real Anal. Exchange, to appear. 
  6. [6] K. Ciesielski, L. Larson and K. Ostaszewski, Density continuity versus continuity, Forum Math. 2 (1990), 265-275. Zbl0714.26002
  7. [7] E. Łazarow, The coarsest topology for I-approximately continuous functions, Comment. Math. Univ. Carolin. 27 (4) (1986), 695-704. Zbl0613.26003
  8. [8] R. J. O'Malley, Baire*1, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187-192. 
  9. [9] W. Poreda, E. Wagner-Bojakowska and W. Wilczyński, A category analogue of the density topology, Fund. Math. 125 (1985), 167-173. Zbl0613.26002
  10. [10] W. Wilczyński, A category analogue of the density topology, approximate continuity and the approximate derivative, Real Anal. Exchange 10 (1984/85), 241-265. Zbl0593.26008

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