View source for Zahorski property

In his fundamental paper [[#References|[a4]]], Z. Zahorski studied (among other topics) zero sets of approximately continuous functions (cf. [[Approximate continuity|Approximate continuity]]; the zero set of a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300201.png" /> is the set of points at which the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300202.png" /> is precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300203.png" />). In modern language, Zahorski proved [[#References|[a4]]], Lemma 11, that given a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300204.png" /> of the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300205.png" />, there is an approximately continuous non-negative bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300206.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300207.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300208.png" /> if and only if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z1300209.png" /> is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002010.png" /> (cf. also [[Set of type F sigma(G delta)|Set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002012.png" />)]]) and closed in the density topology. (Recall that the density topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002013.png" /> is formed by the collection of all Lebesgue measurable sets having each of their points as a [[Density point|density point]].)

Notice that the class of approximately continuous functions was introduced by A. Denjoy in [[#References|[a1]]] as a generalization of the notion of [[Continuity|continuity]]. It is known that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002014.png" /> is approximately continuous if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002015.png" /> is continuous in the density topology. Functions that are approximately continuous have many pleasing properties. For example, they have the [[Darboux property|Darboux property]] and belong to the first Baire class (cf. [[Baire classes|Baire classes]]). Moreover, any bounded approximately continuous function is a [[Derivative|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002017.png" /> is a Pompeiu function if it has a bounded derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002018.png" /> and if the sets on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002019.png" /> is zero or does not vanish, respectively, are both dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002020.png" /> (cf. also [[Dense set|Dense set]]). Also, Zahorski's theorem can serve as a main tool in proving a strengthened form of an old Ward's result from [[#References|[a3]]]: Given a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002021.png" /> of [[Lebesgue measure|Lebesgue measure]] zero, there is an approximately continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002025.png" /> (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002026.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002027.png" />, denote the left-hand upper, respectively right-hand lower, approximative derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002028.png" />).

Consider now a [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002029.png" /> equipped with another topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002030.png" /> which is finer than the original metric topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002031.png" />. The topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002032.png" /> has the Luzin–Menshov property with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002033.png" /> if for each pair of disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002034.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002035.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002036.png" />-closed and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002037.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002038.png" />-closed, there is a pair of disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002039.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002040.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002041.png" />-open and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002042.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002043.png" />-open, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002045.png" />. If the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002046.png" /> has the Luzin–Menshov property with respect to the metric topology, then it has the Zahorski property: Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002047.png" />-closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002048.png" /> which is of the metric type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002049.png" /> is the zero set of a bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002050.png" />-continuous and metric upper [[Semi-continuous function|semi-continuous function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002051.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002052.png" /> 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 [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Denjoy,   "Sur les fonctions dérivées sommables"  ''Bull. Soc. Math. France'' , '''43'''  (1915)  pp. 161–248</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lukeš,   J. Malý,   L. Zajíček,   "Fine topology methods in real analysis and potential theory" , ''Lecture Notes in Mathematics'' , '''1189''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.J. Ward,   "On the points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130020/z13002053.png" />"  ''J. London Math. Soc.'' , '''8'''  (1933)  pp. 293–299</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Z. Zahorski,   "Sur la première dérivée"  ''Trans. Amer. Math. Soc.'' , '''69'''  (1950)  pp. 1–54</TD></TR></table>