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\begin{document}

\noindent Krzysztof Ciesielski, Department of Mathematics, West
Virginia University, Morgantown, WV 26506-6310 (KCies@wvnvms.wvnet.edu)

\medskip

\begin{center}\large\bf Density Topology\end{center}

\bigskip

The density topology $\T_d$ 
on $\real$ is the family 
of all $X\subset\real$ with the property that
every $x\in X$ is a 
density point of $X$, i.e., 
such that 
$\lim_{h\to 0^+}(2h)^{-1}\measure_i(X\cap(x-h,x+h))=1$,
where $\measure_i$ stands for the Lebesgue inner measure. 

The density topology was first defined in~1952 by 
Haupt and Pauc~\cite{HauptPauc:topologie} although its 
study did not start until~1961 
when it was rediscovered by Goffman and Waterman~\cite{GW:AppContTrans}.
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 $\A$ \cite{Denjoy:nombresDerives}.
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 $\real$ are known
quite well. Every $X\in\T_d$ is Lebesgue measurable. 
The topology is connected, completely regular but not normal.
A set $S\subset\real$ is $\T_d$-nowhere dense if and only if 
it has Lebesgue measure zero. 
Also, $\real$ considered with the bi-topological structure~\cite{Ke}
of the density and natural topologies is normal in 
the bi-topological sense. (This is known as the Lusin-Menchoff 
Theorem~\cite{Br}.)


The density topology 
on $\real^n$ for $n\geq 2$ 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 $x\in X\subset\real^2$ satisfying the condition
$\lim_{{\rm diam}(S)\to 0}(\measure_i(S))^{-1}\measure_i(X\cap S)=1$,
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 $\real^2$.
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~\cite{GNN}. However, from the real analysis point of view, the
strong density topology is usually more useful~\cite{Gu}.
%~\cite[pp. 106, 128]{Saks}.


A category analog of the density topology,
introduced by Wilczy\'{n}ski~\cite{Wi},
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~\cite{CLO}.
This monograph contains an 
exhaustive study of sixteen different classes of
continuous functions (from $\real$ to $\real$) 
that can be formed by putting the natural topology or 
either of these density topologies on the domain and the range. 


\begin{thebibliography}{222}

\bibitem[Br]{Br}
A. M. Bruckner,  
{\it Differentiation of Real Functions}, 
Lecture notes in Mathematics {\bf 659},
Springer-Verlag 1978. 

\bibitem[CLO]{CLO}
K. Ciesielski, L. Larson and K. Ostaszewski,  
{\it $\I$-density continuous functions}, Memoirs of the AMS vol. {\bf 107} 
no.~515, 1994.

\bibitem[De]{Denjoy:nombresDerives}
A. Denjoy, {\it M\'{e}moire sur les d\'{e}riv\'{e}s des fonctions continues},
Journ. Math. Pures et Appl. {\bf 1} (1915), 105--240.

\bibitem[GNN]{GNN} C. Goffman, C.J. Neugebauer, T. Nishiura,
{\it Density topology and approximate continuity,}
 Duke Math. J. {\bf 28} (1961), 497--506.

\bibitem[GW]{GW:AppContTrans}
Casper Goffman and Daniel Waterman,
{\it Approximately Continuous Transformations},
Proc. of AMS {\bf 12} (1961), 116--121.

\bibitem[Gu]{Gu} M.~de Guzm\'{a}n,  
{\it Differentiation of Integrals in $\real^n$}, 
Lecture notes in Mathematics {\bf 481},
Springer-Verlag 1975. 

\bibitem[HP]{HauptPauc:topologie}
O. Haupt and Ch. Pauc, 
{\it La topologie de {D}enjoy envisag\'{e}e comme vraie topologie}, 
C. R. Acad. Sci. Paris {\bf 234} (1952), 390--392.

\bibitem[Ke]{Ke} W. C. Kelly,  
{\it Bitopological spaces}, 
Proc. London Math. Soc. {\bf 13} (1963), 71--89.

\bibitem[Wi]{Wi} W. Wilczy\'{n}ski,  
{\it A generalization of the density topology}, 
Real Anal. Exchange {\bf 8} (1982--82), 16--20.

%\bibitem[Sa]{Saks} S. Saks, {\it Theory of the integral},
%Monografie Matematyczne, Warsaw 1937.

\end{thebibliography}

\end{document}