Category theorems concerning $\mathcal I$-density continuous functions

Volume 140 / 1991

Krzysztof Ciesielski, Lee Larson Fundamenta Mathematicae 140 (1991), 79-85 DOI: 10.4064/fm_1991_140_1_1_79_85


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-density} topology 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.


  • Krzysztof CiesielskiDepartment of Mathematics
    West Virginia University
    Morgantown, West Virginia 26506
  • Lee LarsonDepartment of Mathematics
    University of Louisville
    Louisville, Kentucky 40292

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