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Category theorems concerning Z-density continuous functions

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Ciesielski K., Larson L.

Fundamenta Mathematicae

1991-1992

tom 140

nr 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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Ciesielski K.

1 Larson L.

1 1991

Wyniki wyszukiwania

Category theorems concerning Z-density continuous functions