Category theorems concerning $\mathcal I$-density continuous functions
Fundamenta Mathematicae, Tome 140 (1991) no. 1, pp. 79-85
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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.
Keywords:
ℑ-density topology, ℑ-density continuous functions, first category sets
Affiliations des auteurs :
Krzysztof Ciesielski 1 ; Lee Larson 2
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author = {Krzysztof Ciesielski and Lee Larson},
title = {Category theorems concerning $\mathcal I$-density continuous functions},
journal = {Fundamenta Mathematicae},
pages = {79--85},
publisher = {mathdoc},
volume = {140},
number = {1},
year = {1991},
doi = {10.4064/fm_1991_140_1_1_79_85},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm_1991_140_1_1_79_85/}
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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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