Zahorski property

In his fundamental paper [a4], Z. Zahorski studied (among other topics) zero sets of approximately continuous functions (cf. Approximate continuity; the zero set of a real-valued function $f$ is the set of points at which the value of $f$ is precisely $0$). In modern language, Zahorski proved [a4], Lemma 11, that given a subset $Z$ of the real line $\mathbf{R}$, there is an approximately continuous non-negative bounded function $f$ on $\mathbf{R}$ such that $Z = \{ x \in \mathbf{R} : f ( x ) = 0 \}$ if and only if the set $Z$ is of type $G _ { \delta }$ (cf. also Set of type $F _ { \sigma }$ ($G _ { \delta }$)) and closed in the density topology. (Recall that the density topology on $\mathbf{R}$ is formed by the collection of all Lebesgue measurable sets having each of their points as a density point.)

Notice that the class of approximately continuous functions was introduced by A. Denjoy in [a1] as a generalization of the notion of continuity. It is known that a function $f$ is approximately continuous if and only if $f$ is continuous in the density topology. Functions that are approximately continuous have many pleasing properties. For example, they have the Darboux property and belong to the first Baire class (cf. Baire classes). Moreover, any bounded approximately continuous function is a derivative. Hence Zahorski's theorem can be used in constructing functions with peculiar behaviour. For example, it is easy to construct functions of Pompeiu type: A function $f$ on $\mathbf{R}$ is a Pompeiu function if it has a bounded derivative $f ^ { \prime }$ and if the sets on which $f ^ { \prime }$ is zero or does not vanish, respectively, are both dense in $\mathbf{R}$ (cf. also Dense set). Also, Zahorski's theorem can serve as a main tool in proving a strengthened form of an old Ward's result from [a3]: Given a set $E \subset ( 0,1 )$ of Lebesgue measure zero, there is an approximately continuous function $f$ such that $\underline { f } _ { + \text{ap } } = + \infty$ and $\overline { f } _{-\text{ap}} = - \infty$ on $E$ (here, $\overline { f }_{ - \text{ap}}$, respectively $\underline { f } _{+ \text{ap} }$, denote the left-hand upper, respectively right-hand lower, approximative derivative of $f$).

Consider now a metric space $( P , \rho )$ equipped with another topology $\tau$ which is finer than the original metric topology $\tau _ { \rho }$. The topology $\tau$ has the Luzin–Menshov property with respect to $\tau _ { \rho }$ if for each pair of disjoint sets $F , F _ { \tau } \subset P$ with $F$ $\tau _ { \rho }$-closed and $F _ { \tau }$ $\tau$-closed, there is a pair of disjoint sets $G , G _ { \tau } \subset P$ with $G$ $\tau _ { \rho }$-open and $G _ { \tau }$ $\tau$-open, such that $F _ { \tau } \subset G$ and $F \subset G _ { \tau }$. If the topology $\tau$ has the Luzin–Menshov property with respect to the metric topology, then it has the Zahorski property: Any $\tau$-closed subset of $P$ which is of the metric type $G _ { \delta }$ is the zero set of a bounded $\tau$-continuous and metric upper semi-continuous function on $P$. Note that, conversely, the Zahorski property does not imply the Luzin–Menshov property. The density topology on the real line has the Luzin–Menshov property. Therefore it has the Zahorski property. Even very general density topologies, or also fine topologies of potential theory, have the Luzin–Menshov property, hence they have the Zahorski property as well.

The Zahorski property can be introduced also in a very general framework of bitopological spaces. By this one understands a set $X$ equipped with two topologies. If such a bitopological space satisfies the so-called "binormality condition" , it has the Zahorski property.

A detailed study of the Zahorski property and its applications is given in [a2].

References

[a1]	A. Denjoy, "Sur les fonctions dérivées sommables" Bull. Soc. Math. France , 43 (1915) pp. 161–248
[a2]	J. Lukeš, J. Malý, L. Zajíček, "Fine topology methods in real analysis and potential theory" , Lecture Notes in Mathematics , 1189 , Springer (1986)
[a3]	A.J. Ward, "On the points where $A D _ { + } < A D ^ { - }$" J. London Math. Soc. , 8 (1933) pp. 293–299
[a4]	Z. Zahorski, "Sur la première dérivée" Trans. Amer. Math. Soc. , 69 (1950) pp. 1–54