# Applications of Point Set Theory in Real Analysis by A.B. Kharazishvili

By A.B. Kharazishvili

This publication is dedicated to a couple effects from the classical aspect Set thought and their functions to sure difficulties in mathematical research of the true line. become aware of that a number of subject matters from this thought are offered in different books and surveys. From one of the most crucial works dedicated to aspect Set thought, allow us to firstly point out the superb ebook through Oxtoby [83] within which a deep analogy among degree and type is mentioned intimately. additional, an enticing common method of difficulties touching on degree and type is constructed within the recognized monograph via Morgan [79] the place a primary idea of a class base is brought and investigated. We additionally desire to point out that the monograph through Cichon, W«;glorz and the writer [19] has lately been released. In that ebook, sure sessions of subsets of the genuine line are studied and numerous cardinal valued services (characteristics) heavily attached with these sessions are investigated. evidently, the IT-ideal of all Lebesgue degree 0 subsets of the true line and the IT-ideal of all first classification subsets of a similar line are largely studied in [19], and several other fairly new effects touching on this subject are offered. ultimately, it's average to note the following that a few unique units of issues, the so-called singular areas, are thought of within the classi

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**Example text**

Taking account of this property of the Lebesgue measure and applying relation 1), we can conclude that our function tfJ is constant almost everywhere. But, we simultaneously have ra7l(tfJ) S; {O,1}. Thus we get the disjunction: ¢ = 0 almost everywhere or tfJ 1 almost everywhere. But dom( tfJ) R \ Q and the set R \ Q is symmetric with respect to the point O. Now, relation 2) shows that if ifJ = 0 almost everywhere, then ifJ = 1 almost everywhere and, conversely, if ifJ = 1 almost everywhere, then tfJ = 0 almost everywhere.

Let n > be a fixed natural number, let A and B be two sets and let G be an (n - n)-correspondence between these sets. Then there exists a bijection 9 : A --+ B such that the graph of 9 is contained in G. Now, we are going to show that the preceding theorem cannot be proved in the theory ZF & DC. For this purpose, let us return to the Vitali partition {Vi : i E I}. First, let us observe that Q E {Vi : i E I} where Q is the set of all rational numbers. Let us put {Wi : i E l} = {Vi : i E I} \ {Q}.

Now, take the two-element set {O, I} and put A = {Wi : i E I}, B={-WiUWiU{t} : iEI, tE{O,l}}. Furthermore, let us define a binary relation G between the sets A and B. For each element Wi E A, let us put G(Wi) = {-W;UWiU{t} : tE{O,l}}. Obviously, if - Wi U Wi U {t} belongs to the set B, then G- 1 ( -Wi U Wi U {t}) = {-Wi' Wi}. Hence we see that G is a (2 - 2)-correspondence between the sets A and B. We shall show that the existence of a bijection 9 : A --+ B with the graph contained in G cannot be established in the theory ZF & DC.