Geodesic
Category theorems concerning $\mathcal I$-density continuous functions
Krzysztof Ciesielski1 ; Lee Larson2
1Department of Mathematics West Virginia University Morgantown, West Virginia 26506 U.S.A.
2Department of Mathematics University of Louisville Louisville, Kentucky 40292 U.S.A
Fundamenta Mathematicae, Tome 140 (1991) no. 1, pp. 79-85
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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.
DOI : 10.4064/fm_1991_140_1_1_79_85
Keywords: ℑ-density topology, ℑ-density continuous functions, first category sets

Krzysztof Ciesielski  1   ; Lee Larson  2

1 Department of Mathematics West Virginia University Morgantown, West Virginia 26506 U.S.A.
2 Department of Mathematics University of Louisville Louisville, Kentucky 40292 U.S.A 
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Krzysztof Ciesielski; Lee Larson. Category theorems concerning $\mathcal I$-density continuous functions. Fundamenta Mathematicae, Tome 140 (1991) no. 1, pp. 79-85. doi: 10.4064/fm_1991_140_1_1_79_85

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