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Cardinal functions in analysis


The important recent developments in set theoretical analysis concern the cardinal functions that are defined for different classes of real functions. These investigations seem to be analogous to those concerning of cardinal functions in topology from the 1970's and 1980's. (See [71, 67, 72, 139].) They are also related to the deep studies of cardinal invariants associated with different small subsets of the real line. (For a summary of the results concerning cardinals related to the measure and category see [57] or [7]. For a survey concerning cardinals associated with the thin sets derived from harmonic analysis see [17].)

The first group of functions is motivated by the notion of countable continuity and was introduced in 1991 by J. Cichon, M. Morayne, J. Pawlikowski, and S. Solecki in [21]. More precisely, they define the decomposition function tex2html_wrap_inline5983 for arbitrary families tex2html_wrap_inline5985 and tex2html_wrap_inline5987, where tex2html_wrap_inline5989 stands for the set of all functions from X to Y.
where tex2html_wrap_inline5995 denotes the family of all coverings of tex2html_wrap_inline5075 with at most tex2html_wrap_inline5315 many sets. In particular, if tex2html_wrap_inline5827 stands for the family of all continuous functions (from subsets of tex2html_wrap_inline5075 into tex2html_wrap_inline5075) then
In [21] the authors considered the values of tex2html_wrap_inline6011 for tex2html_wrap_inline6013, where tex2html_wrap_inline6015 stands for the functions of tex2html_wrap_inline6017-th Baire class.

The motivation for this definition comes from a question of N. N. Luzin whether every Borel function is countable continuous. This question was answered negatively by P. S. Novikov (see [74]) and was subsequently generalized by Keldys [74] (in 1934), and S. I. Adian and P. S. Novikov [1] (in 1958). The most general result in this direction was obtained in late 1980's by M. Laczkovich [88], who proved, in particular, that tex2html_wrap_inline6019 for every tex2html_wrap_inline6013.

One of the most interesting results from the paper [21] is the following theorem.


It has been also shown by J. Steprans and S. Shelah that none of these inequalities can be replaced by the equation.



There are also some interesting results concerning the value of tex2html_wrap_inline6047, where tex2html_wrap_inline6049 is the class of all (partial) differentiable functions. It has been proved by Morayne [136, Thm 6.1,] that


Also, Steprans proved that


However, the relation between numbers tex2html_wrap_inline6055, tex2html_wrap_inline6057 and tex2html_wrap_inline6011 for tex2html_wrap_inline6061 is unclear.

In the same direction, K. Ciesielski recently noticed that (obviously)
and that it is the best that can be said in ZFC.


In fact, (1) happens in a model obtained by extending a ground model with GCH by adding tex2html_wrap_inline5315 many Cohen reals. The equation tex2html_wrap_inline6081 follows immediately from Theorem 2.9.

The model for (2) is obtained as follows. You start with a model with GCH, assume that tex2html_wrap_inline6083 and take an increasing sequence tex2html_wrap_inline6085 cofinal with tex2html_wrap_inline6087 and such that each tex2html_wrap_inline6089 is a cardinal successor. The desired model is obtained by a generic extension via forcing P which a finite support iteration of forcings tex2html_wrap_inline6093, where each tex2html_wrap_inline6093 is a standard ccc forcing adding the Martin's Axiom over the previous model and making tex2html_wrap_inline6097.

The second group of cardinal functions is defined in terms of algebraic operations on functions. Their definition was motivated by the following property of Darboux functions (from tex2html_wrap_inline5075 to tex2html_wrap_inline5075) due to Fast and mentioned in the previous section:

where |Z| denotes the cardinality of Z. In 1974 Kellum [75] proved the similar result for the class tex2html_wrap_inline5819 of almost continuous functions and in 1991 Natkaniec [104] defined the following cardinal functions for every tex2html_wrap_inline6119 to study these phenomena more closely.
  The extra assumption that tex2html_wrap_inline6129 is added in the definition of tex2html_wrap_inline6131 since otherwise for every family tex2html_wrap_inline6133 containing constant zero function tex2html_wrap_inline6135 we would have tex2html_wrap_inline6137.

Its easy to see that the functions tex2html_wrap_inline6139 and tex2html_wrap_inline6131 are monotone in a sense that tex2html_wrap_inline6143 and tex2html_wrap_inline6145 for every tex2html_wrap_inline6147. Also clearly (1) is false for tex2html_wrap_inline6149. Thus, in language of the function tex2html_wrap_inline6139 Fast and Kellum's results can be expressed as follows:
If tex2html_wrap_inline6153 (so, under the Generalized Continuum Hypothesis GCH) the values of tex2html_wrap_inline6155 and tex2html_wrap_inline6157 are clear: tex2html_wrap_inline6159. Thus, Natkaniec asked [104, p. 495,] (see also [63, Problem 1,]) whether the equation tex2html_wrap_inline6161 can be proved in ZFC.

This question was investigated by Ciesielski and Miller in 1994. They proved that tex2html_wrap_inline6163, that the cofinality tex2html_wrap_inline6165 of tex2html_wrap_inline6157 is greater than tex2html_wrap_inline5125 and that this, together with the inequalities tex2html_wrap_inline6171 is essentially all that can be proved in ZFC.

In particular, Theorem 4.7 says that tex2html_wrap_inline6157 does not have to be a regular cardinal (part (d)) and that tex2html_wrap_inline6157 can be any regular cardinal number between tex2html_wrap_inline6197 and tex2html_wrap_inline6199, with tex2html_wrap_inline6199 being ``arbitrarily large'' (part (c)).

At the same time Natkaniec and Rectex2html_wrap6795 aw established the values of tex2html_wrap_inline6203 and tex2html_wrap_inline6205 proving


The first systematic study of functions tex2html_wrap_inline6139 and tex2html_wrap_inline6131 was done by Ciesielski and Rectex2html_wrap6795 aw in the later part of 1995. They collected basic properties of operators tex2html_wrap_inline6139 and tex2html_wrap_inline6131, which are stated below, and found the values of tex2html_wrap_inline6139 and tex2html_wrap_inline6131 for some other classes of functions.


In particular, (4) from Proposition 4.9 shows that every function is a difference of two functions from a class tex2html_wrap_inline5755 if and only if tex2html_wrap_inline6259.

To state the other results from [41] recall the definitions the following classes of functions, where X is an arbitrary topological space.

tex2html_wrap_inline6263 of connectivity functions tex2html_wrap_inline6265, i.e., such that the graph of f restricted to C (that is tex2html_wrap_inline6271) is connected in tex2html_wrap_inline6273 for every connected subset C of X.
tex2html_wrap_inline6279 of extendable functions tex2html_wrap_inline6265, i.e., such that there exists a connectivity function tex2html_wrap_inline6283 with f(x)=g(x,0) for every tex2html_wrap_inline6287.
tex2html_wrap_inline6289 of peripherally continuous functions tex2html_wrap_inline6265, i.e., such that for every tex2html_wrap_inline6287 and any pair tex2html_wrap_inline6295 and tex2html_wrap_inline6297 of open neighborhoods of x and f(x), respectively, there exists an open neighborhood W of x with tex2html_wrap_inline6307 and tex2html_wrap_inline6309, where tex2html_wrap_inline6311 and tex2html_wrap_inline6313 stand for the closure and the boundary of W, respectively.

We will write tex2html_wrap_inline6317, tex2html_wrap_inline6319 and tex2html_wrap_inline6321 in place of tex2html_wrap_inline6263, tex2html_wrap_inline6279, and tex2html_wrap_inline6289 if tex2html_wrap_inline6329. Notice also, that tex2html_wrap_inline6331 if and only if f is weakly continuous, as defined on page gif.

For the generalized continuity classes of functions (from tex2html_wrap_inline5075 into tex2html_wrap_inline5075) defined so far we have the following proper inclusions tex2html_wrap_inline6339, marked by arrows tex2html_wrap_inline6341. (See [13].)


Chart 1.

In particular, inclusions tex2html_wrap_inline6343, monotonicity of tex2html_wrap_inline6139 and Theorem 4.7(a) imply that tex2html_wrap_inline6347. Similarly, Theorem 4.8 implies that tex2html_wrap_inline6349. The values of tex2html_wrap_inline6139 and tex2html_wrap_inline6131 for the remaining classes are as follows.

Notice also that tex2html_wrap_inline6363. Thus, by monotonicity of tex2html_wrap_inline6139 and the above theorem we obtain the following corollary.


The values of functions tex2html_wrap_inline6139 and tex2html_wrap_inline6131 for the class tex2html_wrap_inline5829 has been studied by Ciesielski and Natkaniec. First they noticed that if the definition of tex2html_wrap_inline6131 from page gif is used then trivially tex2html_wrap_inline6379, since for any function tex2html_wrap_inline6381 with tex2html_wrap_inline6383 for some tex2html_wrap_inline6385 we have tex2html_wrap_inline6387 for every tex2html_wrap_inline6389. Thus, they modified the definition of tex2html_wrap_inline6391 to
(Note that tex2html_wrap_inline6393 is equal to tex2html_wrap_inline6395 as defined on page gif.) With this agreement in place they proved the following result.


However, the following problems remain open.


Another systematic study of operator tex2html_wrap_inline6139 was done by F. Jordan in 1996. In his study, he examined the values of tex2html_wrap_inline6423 where tex2html_wrap_inline6425 and classes tex2html_wrap_inline5755 are chosen from those discussed above. Notice that tex2html_wrap_inline6423 has the following very nice interpretation:
where tex2html_wrap_inline6437. To make this study non-trivial Jordan notes first that the value of tex2html_wrap_inline6439 does not determine the value of tex2html_wrap_inline6423:


This paper [69] contains also the following results.



The importance of the extra assumptions in (4) and (5) of Theorem 4.15 is not clear. In particular, the following problem is still open.


Note also that (4) and (5) of Theorem 4.15, and Theorem 4.12 imply immediately the following corollary.


Finally, the following three classes of functions have been brought to this picture.

of functions tex2html_wrap_inline5367 having the Cantor Intermediate Value Property, i.e., such that for every tex2html_wrap_inline6501 and for each Cantor set K between f(x) and f(y) there is a Cantor set C between x and y such that tex2html_wrap_inline6515;
of functions tex2html_wrap_inline5367 having the Strong Cantor Intermediate Value Property, i.e., such that for every tex2html_wrap_inline6501 and for each Cantor set K between f(x) and f(y) there is a Cantor set C between x and y such that tex2html_wrap_inline6515 and tex2html_wrap_inline5815 is continuous;
of functions tex2html_wrap_inline5367 having the Weak Cantor Intermediate Value Property, i.e., such that for every tex2html_wrap_inline6501 with f(x)<f(y) there is a Cantor set C between x and y such that tex2html_wrap_inline6553.

They fit Chart 1 in the following way. (See Gibson [61].)


Chart 2: ``Darboux like'' functions.

Clearly the above inclusions, monotonicity of tex2html_wrap_inline6139 and tex2html_wrap_inline6131, and Theorem 4.10 imply immediately:
The values of functions tex2html_wrap_inline6139 and tex2html_wrap_inline6131 for the class tex2html_wrap_inline6539, and for the classes formed by the intersections of tex2html_wrap_inline5829 with each of the remaining classes mentioned above were not studied too carefully so far. However, obviously tex2html_wrap_inline6567 implying
Also, it follows from Theorem 3.10 that
while also
A stronger version of this last inequality follows also from the following recent theorem of K. Banaszewski and Natkaniec.

In particular,
This last inequality has been recently improved by F. Jordan, who proved the following.
This theorem gives the value of tex2html_wrap_inline6139 for many classes that can be obtained intersecting classes from Chart 2 and tex2html_wrap_inline5829.

Several other operators similar to tex2html_wrap_inline6139 and tex2html_wrap_inline6131 have also been studied. Thus, in 1995 Natkaniec [107] introduced the following operators connected to the composition of functions, where tex2html_wrap_inline6599 stands for the family of all constant functions.
He proved also the following.


Similar functions have been also studied by Ciesielski and Natkaniec [38]:
where tex2html_wrap_inline6395 (tex2html_wrap_inline6613) is the set of all tex2html_wrap_inline6615 for which there exists tex2html_wrap_inline6617 such that tex2html_wrap_inline6619 (tex2html_wrap_inline6621, respectively). In fact, the class tex2html_wrap_inline6395 has the following nice characterization:
  In [38] the authors proved that


Also, in a recent short survey paper [109] Natkaniec evaluated the values of operators tex2html_wrap_inline6139, tex2html_wrap_inline6131, tex2html_wrap_inline6645 and tex2html_wrap_inline6647 for the class tex2html_wrap_inline6649 of almost continuous functions in sense of Husain, i.e., such tex2html_wrap_inline5367 that tex2html_wrap_inline6653 for every non-empty open set tex2html_wrap_inline6655.


Some other cardinal operators connected with composition and concerning some kind of coding were also studied by Ciesielski and Rectex2html_wrap6795 aw [41], Ciesielski and Natkaniec [38], and Natkaniec [109].

Another variant of function tex2html_wrap_inline6139 is connected to the families of bounded functions. To define it properly the following notation is necessary. For a family tex2html_wrap_inline6671 let tex2html_wrap_inline6673 stand for all uniformly bounded families tex2html_wrap_inline6675, and let tex2html_wrap_inline6677 be the class of all bounded functions tex2html_wrap_inline5367. Then we define
In 1994 Maliszewski [94] proved that
so that tex2html_wrap_inline6683. Moreover, he proved that if all functions in tex2html_wrap_inline5755 are measurable (have Baire property), then we can also assume that the ``universal summand'' bounded function has the same property. Similar results were also proved for families of Borel measurable functions.

The values of tex2html_wrap_inline6687 for the other classes of functions from Chart 1 has been investigated by Ciesielski and Maliszewski [36]. In particular, they proved

Notice also that Theorem 4.22 implies immediately the following corollary.
In particular, Corollary 4.23(1) generalizes a result of Darji and Humke [49] that every bounded function can be expressed a sum of three bounded almost continuous functions. On the other hand Corollary 4.23(2) shows that the following result of Natkaniec is sharp.


It might be also interesting to examine a bounded version of tex2html_wrap_inline6131, defined as
However this function has not been studied so far.

One might also consider the study of the operator tex2html_wrap_inline6139 (and tex2html_wrap_inline6131) for the functions from tex2html_wrap_inline5073 into tex2html_wrap_inline5075 with n>1. This has indeed been done by Ciesielski and Wojciechowski in [44]. The study concerned only the classes tex2html_wrap_inline6713, tex2html_wrap_inline6715, tex2html_wrap_inline6717, tex2html_wrap_inline6719, and tex2html_wrap_inline6721 since other classes from Chart 2 do not have natural generalizations into functions of more than one variable. First, one should recall that for n>1 Chart 1 is not valid any more. The new inclusions (for n>1) are as follows:

(The inclusion ``tex2html_wrap_inline6731'' was proved by Hamilton [66] and by Stallings [132], and the inclusion ``tex2html_wrap_inline6733'' by Hagan [65]. The proof of the inclusion ``tex2html_wrap_inline6735'' is presented in [132]. The examples showing that tex2html_wrap_inline6737 and tex2html_wrap_inline6739 can be found in [105, Examples 1.1.9 and 1.1.10,] or [104, Examples 1.7 and 1.6,], while a simple Baire class 1 function in tex2html_wrap_inline6741 was described in [116, Example 1,].) We do not know whether the inclusion tex2html_wrap_inline6743 is proper.

The problem with studying the value of the operator tex2html_wrap_inline6139 for all these classes (except tex2html_wrap_inline6715) is that there exists a function tex2html_wrap_inline6749 which is not a sum of n Darboux functions, implying that
However, every function function tex2html_wrap_inline5735 is sum of n+1 extendable functions. To express these results nicely, define for tex2html_wrap_inline6757 the repeatability tex2html_wrap_inline6759 of tex2html_wrap_inline5755 as the smallest integer k such that any function tex2html_wrap_inline6765 can be expressed as a sum of k functions from tex2html_wrap_inline5755. (We put tex2html_wrap_inline6771 if such a number does not exist.) In this language the results of Ciesielski and Wojciechowski can be stated as follows.

Clearly Theorem 4.25 implies that tex2html_wrap_inline6773. The problem (stated in [44]) whether this equation can be replaced by the equality has been recently solved by F. Jordan.


The value of tex2html_wrap_inline6781 is clearly equal to 2, since Natkaniec [104] proved that tex2html_wrap_inline6785. This fact has been recently improved by F. Jordan, who proved


Notice also, that in the language of tex2html_wrap_inline6789 operator the results from Theorem 4.24 and Corollary 4.23(2) can be expressed by the equation
where tex2html_wrap_inline6791 is the natural generalization of tex2html_wrap_inline6789 for the class of bounded functions.

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