Density topology


Krzysztof Ciesielski

(Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, U.S.A.)

The density topology T(d) on R is the family of all subsets X of R with the property that every x in X is a Lebesgue density point of X, i.e., such that

measure of X intersected with (x-h,x+h)
---------------------------------------   ----->  1
         measure of  (x-h,x+h)

as h --> 0+.

The density topology was first defined in 1952 by Haupt and Pauc (La topologie de Denjoy envisagee comme vraie topologie, C. R. Acad. Sci. Paris 234 (1952), 390--392) although its study did not start until 1961 when it was rediscovered by Goffman and Waterman (Approximately Continuous Transformations, Proc. of AMS 12 (1961), 116--121). In both cases it was introduced to show that the class A of the approximately continuous functions coincides with the class C(T(d)) 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 Denjoy defined and studied the class A. The equation A=C(T(d)) shows the importance of the density topology in real analysis, since the class A 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 R are known quite well. Every X in T(d) is Lebesgue measurable. The topology is connected, completely regular but not normal. A set S of R is T(d)-nowhere dense if and only if it has Lebesgue measure zero. Also, R considered with the bi-topological structure of the density and natural topologies is normal in the bi-topological sense. (This is known as the Lusin-Menchoff Theorem.)

The density topology on the n-dimensional Euclidean space R^n for n>1 is also defined from the notion of a density point. However, in this case there are different notions of the density point depending of different neighbourhood bases at the point. For example, all points x in X (X sunset of R^2) satisfying the condition

measure of X intersected with S
--------------------------------   ----->  1
         measure of S

as diameter(S) --> 0+

where the sets S are chosen among the squares centered at x, are called ordinary density points of X. This leads to the ordinary density topology on R^2. Similarly, by choosing the sets S from the family of all rectangles centered at x with sides parallel to the axes we obtain the strong density points and strong density topology. The ordinary density topology is completely regular, unlike the strong density topology. However, from the real analysis point of view, the strong density topology is usually more useful.

A category analog of the density topology, introduced by Wilczynski, is called the I-density topology. It is Hausdorff, but not regular. The weak topology generated by the class of all I-approximately continuous functions is known as the deep I-density topology. It is completely regular, but not normal.

Most of the topological information concerning the topologies T(d) and its category analogues can be found in: K. Ciesielski, L. Larson and K. Ostaszewski, I-density continuous functions, Memoirs of the AMS vol. 107 no. 515, 1994. This monograph contains an exhaustive study of sixteen different classes of continuous functions (from R to R) that can be formed by putting the natural topology or either of these density topologies on the domain and the range.