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The density topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101401.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101402.png" /> is the family of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101403.png" /> with the property that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101404.png" /> is a density point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101405.png" />, i.e., such that
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{{MSC|28A05|54A05}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101406.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101407.png" /> stands for the Lebesgue inner measure (cf. [[Lebesgue measure|Lebesgue measure]]).
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{{TEX|done}}
  
The density topology was first defined in 1952 by O. Haupt and Ch. Pauc [[#References|[a7]]], although its study did not start until 1961, when it was rediscovered by C. Goffman and D. Waterman [[#References|[a5]]]. In both cases it was introduced to show that the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101408.png" /> of approximately continuous functions (cf. [[Approximate continuity|Approximate continuity]]) coincides with the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d1101409.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014010.png" /> [[#References|[a3]]]. The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014011.png" /> shows the importance of the density topology in real analysis, since the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014012.png" /> 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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014013.png" /> are known quite well. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014014.png" /> is Lebesgue measurable. The topology is connected, completely regular but not normal (cf. also [[Connected space|Connected space]]; [[Completely-regular space|Completely-regular space]]; [[Normal space|Normal space]]). A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014015.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014016.png" />-nowhere dense if and only if it has Lebesgue measure zero (cf. [[Nowhere-dense set|Nowhere-dense set]]). Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014017.png" /> considered with the bitopological structure [[#References|[a8]]] of the density and natural topologies is normal in the bitopological sense. (This is known as the Luzin–Menshov theorem [[#References|[a1]]].)
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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 density topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014020.png" /> satisfying the condition
+
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 {{Cite|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 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}}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014021.png" /></td> </tr></table>
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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.
  
where the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014022.png" /> are chosen among the squares centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014023.png" />, are called ordinary density points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014024.png" />. This leads to the ordinary density topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014025.png" /> [[#References|[a10]]]. Similarly, by choosing the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014026.png" /> from the family of all rectangles centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014027.png" /> with sides parallel to the axes one obtains the strong density points [[#References|[a10]]] and strong density topology. The ordinary density topology is completely regular, unlike the strong density topology [[#References|[a4]]] (cf. also [[Completely-regular space|Completely-regular space]]). However, from the real analysis point of view, the strong density topology is usually more useful [[#References|[a6]]].
+
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.
 
 
A category analogue of the density topology, introduced by W. Wilczyński [[#References|[a9]]], is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014029.png" />-density topology. It is Hausdorff, but not regular (cf. [[Hausdorff space|Hausdorff space]]; [[Regular space|Regular space]]). The [[Weak topology|weak topology]] generated by the class of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014030.png" />-approximately continuous functions is known as the deep <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014032.png" />-density topology. It is completely regular, but not normal (cf. [[Completely-regular space|Completely-regular space]]; [[Normal space|Normal space]]).
 
 
 
Most of the topological information concerning the topologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014033.png" /> and its category analogues can be found in [[#References|[a2]]]. This monograph contains an exhaustive study of sixteen different classes of continuous functions (from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014034.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014035.png" />) that can be formed by putting the natural topology or either of these density topologies on the domain and the range.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.M. Bruckner,  "Differentiation of real functions" , ''CMR Ser.'' , '''5''' , Amer. Math. Soc.  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Ciesielski,  L. Larson,  K. Ostaszewski,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014036.png" />-density continuous functions" , ''Memoirs'' , '''107''' , Amer. Math. Soc. (1994)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"A. Denjoy,  "Mémoire sur les dérivés des fonctions continues"  ''J. Math. Pures Appl.'' , '''1'''  (1915)  pp. 105–240</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"C. Goffman,  C.J. Neugebauer,  T. Nishiura,  "Density topology and approximate continuity"  ''Duke Math. J.'' , '''28'''  (1961)  pp. 497–506</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> C. Goffman,  D. Waterman,  "Approximately continuous transformations" ''Proc. Amer. Math. Soc.'' , '''12'''  (1961)  pp. 116–121</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. de Guzmán,  "Differentiation of integrals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110140/d11014037.png" />" , ''Lecture Notes in Mathematics'' , '''481''' , Springer  (1975)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"O. Haupt,  Ch. Pauc,  "La topologie de Denjoy envisagée comme vraie topologie"  ''C.R. Acad. Sci. Paris'' , '''234'''  (1952)  pp. 390–392</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W.C. Kelly,  "Bitopological spaces"  ''Proc. London Math. Soc.'' , '''13'''  (1963)  pp. 71–89</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> W. Wilczyński,  "A generalization of the density topology"  ''Real Anal. Exchange'' , '''8'''  (1982–82)  pp. 16–20</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , ''Monografie Mat.'' , PWN  (1937)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Br}} A.M. Bruckner,  "Differentiation of real functions" , ''CMR Ser.'' , '''5''' , Amer. Math. Soc.  (1994) {{MR|1274044}}  {{ZBL|0796.26004}}
 +
|-
 +
|valign="top"|{{Ref|CLO}} K. Ciesielski,  L. Larson,  K. Ostaszewski, $\mathcal{I}$-density continuous functions" , ''Memoirs'' , '''107''' , Amer. Math. Soc.   (1994) {{MR|MR1188595}} {{ZBL|0801.26001}}
 +
|-
 +
|valign="top"|{{Ref|Den}} A. Denjoy,  "Mémoire sur les dérivés des fonctions continues"  ''J. Math. Pures Appl.'' , '''1'''  (1915)  pp. 105–24  JFM {{ZBL|45.1285.01}} JFM {{ZBL|45.0435.04}}
 +
|-
 +
|valign="top"|{{Ref|GNN}} C. Goffman,  C.J. Neugebauer,  T. Nishiura,  "Density topology and approximate continuity"  ''Duke Math. J.'' , '''28'''  (1961)  pp. 497–506 {{MR|0137805}}  {{ZBL|0101.15502}}
 +
|-
 +
|valign="top"|{{Ref|GW}} C. Goffman,  D. Waterman,  "Approximately continuous transformations"   ''Proc. Amer. Math. Soc.'' , '''12'''  (1961)  pp. 116–121 {{MR|0120327}}  {{ZBL|0096.17103}}
 +
|-
 +
|valign="top"|{{Ref|deG}} M. de Guzmán,  "Differentiation of integrals in $\mathbb R^n$" , ''Lecture Notes in Mathematics'' , '''481''' , Springer  (1975) {{MR|0457661}}  {{ZBL|0327.26010}}
 +
|-
 +
|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   {{ZBL|0046.05601}}
 +
|-
 +
|valign="top"|{{Ref|Kel}} W.C. Kelly,  "Bitopological spaces"  ''Proc. London Math. Soc.'' , '''13'''  (1963)  pp. 71–89 {{MR|0143169}}  {{ZBL|0107.16401}}
 +
|-
 +
|valign="top"|{{Ref|Sak}} S. Saks,  "Theory of the integral" , ''Monografie Mat.'' , PWN  (1937)  {{MR|0167578}}  {{ZBL|0017.30004}} JFM {{ZBL|63.0183.05}}
 +
|-
 +
|valign="top"|{{Ref|Wil}} W. Wilczyński,  "A generalization of the density topology"  ''Real Anal. Exchange'' , '''8'''  (1982–82)  pp. 16–20   
 +
|-
 +
|}

Latest revision as of 22:40, 17 August 2013

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) MR1274044 Zbl 0796.26004
[CLO] K. Ciesielski, L. Larson, K. Ostaszewski, $\mathcal{I}$-density continuous functions" , Memoirs , 107 , Amer. Math. Soc. (1994) MRMR1188595 Zbl 0801.26001
[Den] A. Denjoy, "Mémoire sur les dérivés des fonctions continues" J. Math. Pures Appl. , 1 (1915) pp. 105–24 JFM Zbl 45.1285.01 JFM Zbl 45.0435.04
[GNN] C. Goffman, C.J. Neugebauer, T. Nishiura, "Density topology and approximate continuity" Duke Math. J. , 28 (1961) pp. 497–506 MR0137805 Zbl 0101.15502
[GW] C. Goffman, D. Waterman, "Approximately continuous transformations" Proc. Amer. Math. Soc. , 12 (1961) pp. 116–121 MR0120327 Zbl 0096.17103
[deG] M. de Guzmán, "Differentiation of integrals in $\mathbb R^n$" , Lecture Notes in Mathematics , 481 , Springer (1975) MR0457661 Zbl 0327.26010
[HP] O. Haupt, Ch. Pauc, "La topologie de Denjoy envisagée comme vraie topologie" C.R. Acad. Sci. Paris , 234 (1952) pp. 390–392 Zbl 0046.05601
[Kel] W.C. Kelly, "Bitopological spaces" Proc. London Math. Soc. , 13 (1963) pp. 71–89 MR0143169 Zbl 0107.16401
[Sak] S. Saks, "Theory of the integral" , Monografie Mat. , PWN (1937) MR0167578 Zbl 0017.30004 JFM Zbl 63.0183.05
[Wil] W. Wilczyński, "A generalization of the density topology" Real Anal. Exchange , 8 (1982–82) pp. 16–20
How to Cite This Entry:
Density topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Density_topology&oldid=13331
This article was adapted from an original article by K. Ciesielski (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article