Some elements of topology

Let X and Y be arbitrary sets. For arbitrary families and , where stands for the collection of all subsets of a set Z, define

and

If families and are the topologies on X and Y, respectively, then is a well known object: the class of all continuous functions from to . Similarly a class of measurable functions with respect to an algebra of subsets of X is equal to , where is an appropriate topology on Y.

In both these approaches one starts with families of sets and and obtain, in return, a family of functions. But what if a class of functions is given to begin with? When can we find families and such that or ? 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 can be

• characterized by images of sets if for some families and ;
• characterized by preimages of sets if for some families and ;
• topologized if for some topologies on X and on Y;
• characterized by associated sets if and for some family and .

From all these notions only the problem of characterizing by associated sets has been extensively studied. Clearly, all classes of continuous function from a topological space X into (considered with the natural topology) can be characterized by associated sets. So can be the family of -measurable functions from X into , for any -algebra 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 to ) in that way. Today we know that derivatives cannot be characterized by associated sets: any class that can be characterized that way has the property that for every and every homeomorphism ; 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 to ) cannot be characterized by associated sets: (Bruckner [14, 1967,]), (B. Cristian, I. Tevy [19, 1980,]), (Kellum [76, 1982,]), (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]. He starts with the following theorem listing basic properties of classes that can be topologized. In the theorem stands for the set of complex numbers, for the class of linear functions f(x)=ax+b, for the natural topology on , and 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 and 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 -ideal of the first category subsets of , and using the fact that for any different harmonic functions we have we can conclude that the class of all harmonic functions can be topologized.

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

 

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

 

Note that a family from Theorem 5.14 is just the family 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 , a Cook continuum, for which , and so, it can be characterized by images of sets.

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

 

This, in particular, implies the following corollary.

 

They also noticed that the class 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 , and of functions (from to ) 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 (-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 , where 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 that were most studied in this aspect in recent years are the -density topology (defined in 1982 by Wilczynski [141]) and the deep -density topologies (defined in 1986 independently by azarow [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.)