ZESZYTY N A W O W E WYŻSZEJ SZKOŁY PEDAGOGICZNEJ v BYDGOSZCZY Problemy Matematyczne 1985 z. 7
MATERIAŁY Z MIĘDZYNARODOWEJ KONFERENCJI TEORII FUNKCJI RZECZYWISTYCH
MAREK BALCERZAK Uniwersytet Łódzki
A GENERALIZATION OF THE THEOREM OF MAULDIN
Let X be a metric space and let ^ be a proper lY-ideal of subsets of X. It will be assumed that all singletons ^x} , xfi. X , belong to 9 .
Denote by the family of all real-valued functions defined on X whose set of points of discontinuity belongs to 3 . For each ordinal <*. , 0 t ф « ) be the family of all pointvise limits of sequences which terms are taken form . The first number oi. such that ф ^ ) = Vill be called the Baire order of the ö'-ideal 3 .
The generalized Baire classes ф (.3) were considered by Mauldin (see
C6J,C73,[8]).
In [2] Kuratowski proved that if X complete and sepa rable, and *3 denotes the <T-ideal of all sets of the first category, then the order of ^ is 1 . In [
7
] Mauldin proved that if 3 denotes the <T-ideal of all subsets of Г о , ij o f the Lebesgue measure zero, then the order of -3 is UJ1 .We have obtained the following generalization of this result: Theorem 1. Let X be a perfect metric space, complete and separable. Let be a -ideal of subsets of X such that
(1) there is a compact set X 'S X such that XQ $ r (2) for each countable set A £ X there is a G r set В
о such that A S B ć J o .
Then for each tf-ideal 3 such that 3 ê tï the order of 3 is £J1 . Remarks and problems, (a) In the case when X =fO,lJ and 3 = 3 is the ideal of sets of the measure zero, we obtain
1 1 8
Mauldin's result.
(b) The condition (1) is fulfilled when X Is looally compact. Indeed, then we put as Xq a compact set which Is a closure of an open nonempty set . Can the condition Cl ) be omitted in the general case?
(c) In [9j Myciel s k i constructed a cT-ideal of subsets t
of the Cantor set С which satisfies the condition (2) . Since X = С is compact, the condition (1) also holds. So Theorem 1 can be applied.
(d) Let X be such as in Theorem 1 and moreover let X be locally compact. Suppose that 9 is a 6"-ideal of X with
the order .
Does there ЛГ-ideal exist such that Sé'J and *3
О о о
fulfils the condition (2) ?
The proof of Theorem 1 is based on the method presented by Mauldin. A new element of the proof is the application of the topology $ (3) associated with the ideal 3 . This topology was investigated by many authors (comp. [1 J ,[ 1*1, [
5
З , Г9
З) . New properties o f t (^) which were used in the proof of Theorem 1 will be presented here.Definition 1. For A t X let A ^ be the set of all X X such that V f \ A ^ 3 for every neighbourhood V of x,
In turns out that A — A ^ s a t i s f i e s all conditions of the operator of the derived set and it yields a topology C(3) .
Definition 2. A closed set A,0 0 A S X will be called 3 -perfect if and only if for eaoh set V such that V f\ A 0 0»
we have V f\ A if 3 .
Proposition 1. A set A, 0 ^ A ç X is -perfect if and only if A = A ^ .
(it means that 1 -perfect sets coincide with perfect sets in the topology Vfi)) »
Proposition 2. For each closed set A S X there is a unique decomposition A = B U С into dijoint sets B,C such that BfeîJ , and С = 0 or С is tJ -perfect,
11 9
If denote* the S'-ideal of all countable seta, then we have the olaaeic formulation. A alnllar result waa obtained by Louveau in fl J •
REFERENCES
[ 1 ] HashimotoH. , On the*topelogy and its application, Fund. Math. 91 (1976) 5-10
[2] Kuratowaki K . , Sur lea fonctions representable analytique-ment et lea ensembles de premiere catégorie, Fund. Math. 5
(1924), 75-86
[
3
] Louveau A., C -idéaux engendrés par dea ensenbles fermes et théorems d 'approximation, Trana. Amer. Math. Soc. 257(
1980
), 143-169[4 ] Marczewski E., Traczyk T., On developable aeta ard almost -lim its points , Colloq. Math 8 (1961), 55-66
Г
5
] Martin N .F.G., Generalized condensation pointa, Duke Math. J. 28 (1961) , 507-514[6] Mauldin R.D., с -ideała and related Baire sy stem s , Fund. Math. 71 (1
971
), 171-177[7] Mauldin R.D . , The Baire order of the functions continuous almost everywhere, Proc . Amer. Math. Soc. 41 (1973), 535- -540
[8] Mauldin R.D., The Baire order of the function oontinuous almost everywhere II, Proo. Amer. Math. Soc. 51 (1975), 371-377
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