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Some elements of topology

Let X and Y be arbitrary sets. For arbitrary families tex2html_wrap_inline6833 and tex2html_wrap_inline6835, where tex2html_wrap_inline6837 stands for the collection of all subsets of a set Z, define
If families tex2html_wrap_inline6841 and tex2html_wrap_inline5493 are the topologies on X and Y, respectively, then tex2html_wrap_inline6849 is a well known object: the class of all continuous functions from tex2html_wrap_inline6851 to tex2html_wrap_inline6853. Similarly a class of measurable functions with respect to an algebra tex2html_wrap_inline6841 of subsets of X is equal to tex2html_wrap_inline6849, where tex2html_wrap_inline5493 is an appropriate topology on Y.

In both these approaches one starts with families of sets tex2html_wrap_inline6841 and tex2html_wrap_inline5493 and obtain, in return, a family of functions. But what if a class of functions tex2html_wrap_inline6869 is given to begin with? When can we find families tex2html_wrap_inline6833 and tex2html_wrap_inline6835 such that tex2html_wrap_inline6875 or tex2html_wrap_inline6877? And how nice can these families be, if they exist?

This questions have been studied recently by several authors. To talk about their results, let us fix the following terminologies. We say that a family tex2html_wrap_inline6869 can be

From all these notions only the problem of characterizing by associated sets has been extensively studied. Clearly, all classes of continuous function tex2html_wrap_inline6911 from a topological space X into tex2html_wrap_inline5075 (considered with the natural topology) can be characterized by associated sets. So can be the family of tex2html_wrap_inline5493-measurable functions from X into tex2html_wrap_inline5075, for any tex2html_wrap_inline6923-algebra tex2html_wrap_inline5493 of subsets of X. However, there are also many examples of classes of functions that do not admit such a characterization. In fact, the real interest in the characterizations of functions by associated sets has been initiated by the 1950 paper of Zahorski [142], in which he tried to characterize derivatives (from tex2html_wrap_inline5075 to tex2html_wrap_inline5075) in that way.gif Today we know that derivatives cannot be characterized by associated sets: any class tex2html_wrap_inline5755 that can be characterized that way has the property that tex2html_wrap_inline6935 for every tex2html_wrap_inline6937 and every homeomorphism tex2html_wrap_inline6939; however derivatives do not have this property. (See Bruckner's book [15] on this subject. Compare also [16].) This negative result has been followed by several others, in which the authors prove that the following classes of functions (from tex2html_wrap_inline5075 to tex2html_wrap_inline5075) cannot be characterized by associated sets: tex2html_wrap_inline5745 (Bruckner [14, 1967,]), tex2html_wrap_inline6317 (B. Cristian, I. Tevy [19, 1980,]), tex2html_wrap_inline5819 (Kellum [76, 1982,]), tex2html_wrap_inline6319 (Rosen [115, 1996,]) and the remaining classes from Chart 2 (Ciesielski, Natkaniec [39, 1997,]).

The question about topologizing different classes of real functions has been first systematically studied in early 1990's by Ciesielski in [26].gif He starts with the following theorem listing basic properties of classes that can be topologized. In the theorem tex2html_wrap_inline6953 stands for the set of complex numbers, tex2html_wrap_inline6955 for the class of linear functions f(x)=ax+b, tex2html_wrap_inline6959 for the natural topology on tex2html_wrap_inline5075, and tex2html_wrap_inline6963 for the identity function from X to X.

Of all these properties only (iii) needs a little longer (but still easy) argument. Note also, that (i) shows, that in order to topologize some family, only the search for the range topology is essential. Condition (v) shows that the question when topologies tex2html_wrap_inline7069 and tex2html_wrap_inline7071 can be chosen equal is answered by the following corollary.


Next, from Theorem 5.1 (conditions (iii), (vi) and (ix)) Ciesielski concludes the following fact


which easily leads to the following corollary:


(The definitions of all classes of functions from this, and the next corollary can be found in [15] and in [35].)

With a little more effort he also concludes


From the positive side, paper [26] contains the following deeper result.


Applying Theorem 5.6 to the tex2html_wrap_inline6923-ideal tex2html_wrap_inline7151 of the first category subsets of tex2html_wrap_inline7153, and using the fact that for any different harmonic functions tex2html_wrap_inline7155 we have tex2html_wrap_inline7157 we can conclude that the class of all harmonic functions tex2html_wrap_inline7159 can be topologized.

Another tex2html_wrap_inline6923-ideal that can be used with Theorem 5.6 is the ideal tex2html_wrap_inline7163 of at most countable sets. Since for any two different analytic functions tex2html_wrap_inline7165 we have tex2html_wrap_inline7167, we can also conclude the following corollary.


Notice also, that if the family tex2html_wrap_inline5755 in Corollary 5.7 is closed under the composition and tex2html_wrap_inline7187, then, by Theorem 5.1(v), tex2html_wrap_inline7189. We can write this in the form of next corollary, where tex2html_wrap_inline6841 stands for the family of all analytic functions and tex2html_wrap_inline7193 for the family of all polynomials.


The following questions in these subject are open.


The general problem of characterizing classes of functions by preimages of sets (in a sense defined above) has been studied only in two papers: [29] and [39]. In paper [29] Ciesielski proves the following theorem, which generalizes a similar result of Preiss and Tartaglia [113].


Clearly the family tex2html_wrap_inline7243 of all derivatives satisfies the above conditions (1)-(3). In particular, Theorem 5.9 implies the following two corollaries.



Note that by Corollary 5.5 the families tex2html_wrap_inline5745 and tex2html_wrap_inline6841 in Corollary 5.11 cannot be topologies. Also, they cannot be algebras:


The following problems remain open.


The problem of characterizing by preimages of sets families from Chart 2 has been recently addressed by Ciesielski and Natkaniec.


The problem of characterizing a family of functions by images of sets was first studied by Velleman for the class tex2html_wrap_inline5827 of continuous functions from tex2html_wrap_inline5075 to tex2html_wrap_inline5075.


Note that a family tex2html_wrap_inline7347 from Theorem 5.14 is just the family tex2html_wrap_inline5745 of Darboux functions.

Theorem 5.14(2) has been essentially generalized by Ciesielski, Dikranjan and Watson in [30]. In this paper the authors list a basic properties of classes that can be characterized by images of sets, which is similar in flavor to Theorem 5.1. Then, they prove the following generalization of Theorem 5.14.


They also remarked that there is a compact subset tex2html_wrap_inline7361, a Cook continuum, for which tex2html_wrap_inline7363, and so, it can be characterized by images of sets.

For the classes of functions from tex2html_wrap_inline5075 to tex2html_wrap_inline5075, their generalization of Theorem 5.14 appears as follows.


This, in particular, implies the following corollary.


They also noticed that the class tex2html_wrap_inline5745 of Darboux functions can be characterized by images of sets. (It is defined that way.)

It has been also recently noticed by Ciesielski and Natkaniec [39] that in Theorem 5.15 the clause ``non-measurable'' cannot be replaced by ``without the Baire property.'' More precisely, they proved


Finally, Ciesielski and Natkaniec [39] proved that it is impossible to characterize by images of sets the classes tex2html_wrap_inline5829, and tex2html_wrap_inline7403 of functions (from tex2html_wrap_inline5075 to tex2html_wrap_inline5075) with the Baire property. They also proved the following theorem.


The following problem in this area remain open.


Another interesting problem (loosely related to real functions, but having the same flavor that the topologizing question has) concerns the existence of a topology on a given set X, often the real line, satisfying the best possible separation axioms, for which a given ideal (tex2html_wrap_inline6923-ideal) of subsets of X consists precisely of sets that are nowhere dense (or first category) in X. Ciesielski and Jasinski [31, 1995,] obtained several positive results in this direction under some additional set-theoretic assumptions. The problem was also investigated in the papers [114] by Rogowska and [4] by Balcerzak and Rogowska.

There are also many interesting theorems concerning different classes of functions tex2html_wrap_inline6749, where tex2html_wrap_inline5741 is equipped with some abstract topology refining of the natural topology. A survey of some recent results in this direction can be found in the last issue of the Real Analysis Exchange [64]. The topologies on tex2html_wrap_inline5075 that were most studied in this aspect in recent years are the tex2html_wrap_inline7441-density topology (defined in 1982 by Wilczynski [141]) and the deep tex2html_wrap_inline7441-density topologies (defined in 1986 independently by tex2html_wrap7445azarow [91], and by Poreda and Wagner-Bojakowska [112]). These are category analogues of the density topology. The survey of the results in this direction can be found in a monograph of Ciesielski, Larson and Ostaszewski [35]. (In particular, see [32] or [35, Sec. 1.5,] for some set theoretic results and open problems concerning these topologies.)

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Next: Elements of measure theory Up: Set Theoretic Real Analysis Previous: Cardinal functions in analysis