Density topology

2020 Mathematics Subject Classification: Primary: 28A05 Secondary: 54A05 [MSN][ZBL]

The density topology $\mathcal{T}_d$ on $\mathbb R$ consists of the family of all subsets $E\subset \mathbb R$ with the property that every $x\in E$ has density $1$ with respect to the Lebesgue measure $\lambda$, that is \[ \lim_{\delta\downarrow 0} \frac{\lambda (E\cap ]x-\delta, x+\delta[)}{2\delta} = 1 \qquad \forall x\in E\, . \] The density topology was first defined in 1952 by O. Haupt and Ch. Pauc [HP], although its study did not start until 1961, when it was rediscovered by C. Goffman and D. Waterman [GW]. In both cases it was introduced to show that the class $\mathcal{A}$ of approximately continuous functions coincides with the class $C (\mathcal{T}_d)$ 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 A. Denjoy defined and studied the class $\mathcal{A}$ [Den]. The equation $\mathcal{A} = C (\mathcal{T}_d)$ shows the importance of the density topology in real analysis, since the class $\mathcal{A}$ 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 $\mathbb R$ are known quite well. Every $E\in \mathcal{T}_d$ is Lebesgue measurable. The topology is connected, completely regular but not normal. A set $E\subset \mathbb R$ is $\mathcal{T}_d$-nowhere dense if and only if it has Lebesgue measure zero. Also, $\mathbb R$ considered with the bitopological structure [Kel] of the density and natural topologies is normal in the bitopological sense. (This is known as the Luzin–Menshov theorem [Br].)

The concept can be extended to the higher dimensional case in more than one way, depending on the class of neighborhoods used to determine the notion of density. The standard notion of ordinary density points leads to the ordinary density topology, that is the class $\mathcal{T}_d^o$ of sets $E\subset \mathbb R^n$ for which \[ \lim_{r\downarrow 0} \frac{\lambda (E\cap B_r (x))}{\lambda (B_r (x))} = 1 \qquad \forall x\in E\, . \] Similarly, if we introduce the family $\mathcal{N}_x$ of all rectangles centred at $x$ with sides parallel to the axes, one obtains the strong density points [Sak] and the strong density topology $\mathcal{T}^s_d$ of sets $E$ such that \[ \lim_{R\in \mathcal{N}_x, \text{diam}\, (R)\to 0} \frac{\lambda (R\cap E)}{\lambda (R)} =1 \qquad \forall x\in E\, . \] The ordinary density topology is completely regular, unlike the strong density topology [GNN]. However, from the real analysis point of view, the strong density topology is usually more useful [deG].

A category analogue of the density topology, introduced by W. Wilczyński [Wil], is called the $\mathcal{I}$-density topology. It is Hausdorff, but not regular. The weak topology generated by the class of all $\mathcal{I}$-approximately continuous functions is known as the deep $\mathcal{I}$-density topology. It is completely regular, but not normal.

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

References

[Br] A.M. Bruckner, "Differentiation of real functions" , CMR Ser. , 5 , Amer. Math. Soc. (1994)

[CLO] K. Ciesielski, L. Larson, K. Ostaszewski, $\mathcal{I}$-density continuous functions" , Memoirs , 107 , Amer. Math. Soc. (1994)

[Den] A. Denjoy, "Mémoire sur les dérivés des fonctions continues" J. Math. Pures Appl. , 1 (1915) pp. 105–24

[GNN] C. Goffman, C.J. Neugebauer, T. Nishiura, "Density topology and approximate continuity" Duke Math. J. , 28 (1961) pp. 497–506

[GW] C. Goffman, D. Waterman, "Approximately continuous transformations" Proc. Amer. Math. Soc. , 12 (1961) pp. 116–121

[deG] M. de Guzmán, "Differentiation of integrals in $\mathbb R^n$" , Lecture Notes in Mathematics , 481 , Springer (1975)

[HP] O. Haupt, Ch. Pauc, "La topologie de Denjoy envisagée comme vraie topologie" C.R. Acad. Sci. Paris , 234 (1952) pp. 390–392

[Kel] W.C. Kelly, "Bitopological spaces" Proc. London Math. Soc. , 13 (1963) pp. 71–89

[Sak] S. Saks, "Theory of the integral" , Monografie Mat. , PWN (1937)

[Wil] W. Wilczyński, "A generalization of the density topology" Real Anal. Exchange , 8 (1982–82) pp. 16–20