View source for Density topology

{{MSC|28A05|54A05}}

[[Category:Classical measure theory]]

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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 of a set|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 {{Cite|HP}}, although its study did not start until 1961, when it was rediscovered by C. Goffman and D. Waterman {{Cite|GW}}. In both cases it was introduced to show that the class $\mathcal{A}$ of [[Approximate continuity|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 $\mathcal{A}$ {{Cite|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|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 space|connected]], [[Completely-regular space|completely regular]] but not [[Normal space|normal]]. A set $E\subset \mathbb R$ is $\mathcal{T}_d$-[[Nowhere-dense set|nowhere dense]] if and only if it has Lebesgue measure zero. Also, $\mathbb R$ considered with the bitopological structure {{Cite|Kel}} of the density and natural topologies is normal in the bitopological sense. (This is known as the Luzin–Menshov theorem {{Cite|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 of a set|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 [[#References|[a10]]] 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 space|completely regular]], unlike the strong density topology {{Cite|GNN}}. However, from the real analysis point of view, the strong density topology is usually more useful {{Cite|deG}}.

A category analogue of the density topology, introduced by W. Wilczyński {{Cite|Wil}}, is called the $\mathcal{I}$-density topology. It is [[Hausdorff space|Hausdorff]], but not [[Regular space|regular]]. The [[Weak topology|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 {{Cite|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====
{|
|-
|valign="top"|{{Ref|Br}}  A.M. Bruckner,   "Differentiation of real functions" , ''CMR Ser.'' , '''5''' , Amer. Math. Soc.  (1994)
|-
|valign="top"|{{Ref|CLO}} K. Ciesielski,   L. Larson,   K. Ostaszewski, $\mathcal{I}$-density  continuous functions" , ''Memoirs'' , '''107''' , Amer. Math. Soc.   (1994)
|-
|valign="top"|{{Ref|Den}} A. Denjoy,   "Mémoire sur les dérivés des fonctions continues"  ''J. Math. Pures Appl.'' , '''1'''  (1915)  pp. 105–24
|-
|valign="top"|{{Ref|GNN}} C. Goffman,   C.J. Neugebauer,   T. Nishiura,   "Density topology and  approximate continuity"  ''Duke Math. J.'' , '''28'''  (1961)  pp.  497–506
|-
|valign="top"|{{Ref|GW}}  C. Goffman,   D. Waterman,   "Approximately continuous transformations"   ''Proc. Amer. Math. Soc.'' , '''12'''  (1961)  pp. 116–121
|-
|valign="top"|{{Ref|deG}} M. de Guzmán,   "Differentiation of integrals in $\mathbb R^n$" , ''Lecture Notes in Mathematics'' , '''481''' , Springer  (1975)
|-
|valign="top"|{{Ref|HP}} O. Haupt,   Ch. Pauc,   "La topologie de Denjoy envisagée comme vraie  topologie"  ''C.R. Acad. Sci. Paris'' , '''234'''  (1952)  pp. 390–392
|-
|valign="top"|{{Ref|Kel}} W.C. Kelly,   "Bitopological spaces"  ''Proc. London Math. Soc.'' , '''13'''  (1963)  pp. 71–89
|-
|valign="top"|{{Ref|Sak}} S. Saks,   "Theory of the integral" , ''Monografie Mat.'' , PWN  (1937)
|-
|valign="top"|{{Ref|Wil}}  W. Wilczyński,   "A generalization of the density topology"  ''Real Anal. Exchange'' , '''8'''  (1982–82)  pp. 16–20
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