Density topology

The density topology on is the family of all with the property that every is a density point of , i.e., such that

where stands for the Lebesgue inner measure (cf. Lebesgue measure).

The density topology was first defined in 1952 by O. Haupt and Ch. Pauc [a7], although its study did not start until 1961, when it was rediscovered by C. Goffman and D. Waterman [a5]. In both cases it was introduced to show that the class of approximately continuous functions (cf. Approximate continuity) coincides with the class of all real functions that are continuous with respect to the density topology on the domain and the natural topology on the range. Thus, in a way, the density topology has been present in real analysis since 1915, when A. Denjoy defined and studied the class [a3]. The equation shows the importance of the density topology in real analysis, since the class is strongly tied to the theory of Lebesgue integration and differentiation. For example, a bounded function is approximately continuous if and only if it is a derivative.

The topological properties of the density topology on are known quite well. Every is Lebesgue measurable. The topology is connected, completely regular but not normal (cf. also Connected space; Completely-regular space; Normal space). A set is -nowhere dense if and only if it has Lebesgue measure zero (cf. Nowhere-dense set). Also, considered with the bitopological structure [a8] of the density and natural topologies is normal in the bitopological sense. (This is known as the Luzin–Menshov theorem [a1].)

The density topology on for is also defined from the notion of a density point. However, in this case there are different notions of the density point, depending on different neighbourhood bases at the point. For example, all points satisfying the condition

where the sets are chosen among the squares centred at , are called ordinary density points of . This leads to the ordinary density topology on [a10]. Similarly, by choosing the sets from the family of all rectangles centred at with sides parallel to the axes one obtains the strong density points [a10] and strong density topology. The ordinary density topology is completely regular, unlike the strong density topology [a4] (cf. also Completely-regular space). However, from the real analysis point of view, the strong density topology is usually more useful [a6].

A category analogue of the density topology, introduced by W. Wilczyński [a9], is called the -density topology. It is Hausdorff, but not regular (cf. Hausdorff space; Regular space). The weak topology generated by the class of all -approximately continuous functions is known as the deep -density topology. It is completely regular, but not normal (cf. Completely-regular space; Normal space).

Most of the topological information concerning the topologies and its category analogues can be found in [a2]. This monograph contains an exhaustive study of sixteen different classes of continuous functions (from to ) that can be formed by putting the natural topology or either of these density topologies on the domain and the range.

References

[a1]	A.M. Bruckner, "Differentiation of real functions" , CMR Ser. , 5 , Amer. Math. Soc. (1994)
[a2]	K. Ciesielski, L. Larson, K. Ostaszewski, "-density continuous functions" , Memoirs , 107 , Amer. Math. Soc. (1994)
[a3]	A. Denjoy, "Mémoire sur les dérivés des fonctions continues" J. Math. Pures Appl. , 1 (1915) pp. 105–240
[a4]	C. Goffman, C.J. Neugebauer, T. Nishiura, "Density topology and approximate continuity" Duke Math. J. , 28 (1961) pp. 497–506
[a5]	C. Goffman, D. Waterman, "Approximately continuous transformations" Proc. Amer. Math. Soc. , 12 (1961) pp. 116–121
[a6]	M. de Guzmán, "Differentiation of integrals in " , Lecture Notes in Mathematics , 481 , Springer (1975)
[a7]	O. Haupt, Ch. Pauc, "La topologie de Denjoy envisagée comme vraie topologie" C.R. Acad. Sci. Paris , 234 (1952) pp. 390–392
[a8]	W.C. Kelly, "Bitopological spaces" Proc. London Math. Soc. , 13 (1963) pp. 71–89
[a9]	W. Wilczyński, "A generalization of the density topology" Real Anal. Exchange , 8 (1982–82) pp. 16–20
[a10]	S. Saks, "Theory of the integral" , Monografie Mat. , PWN (1937)