Borel Liftings of Borel Sets: Some Decidable and Undecidable by Gabriel Debs

By Gabriel Debs

One of many goals of this paintings is to enquire a few average houses of Borel units that are undecidable in $ZFC$. The authors' place to begin is the subsequent straightforward, even though non-trivial end result: think of $X \subset 2omega\times2omega$, set $Y=\pi(X)$, the place $\pi$ denotes the canonical projection of $2omega\times2omega$ onto the 1st issue, and believe that $(\star)$ : ""Any compact subset of $Y$ is the projection of a few compact subset of $X$"". If additionally $X$ is $\mathbf{\Pi zero 2$ then $(\star\star)$: ""The limit of $\pi$ to a couple quite closed subset of $X$ is ideal onto $Y$"" it follows that during the current case $Y$ is additionally $\mathbf{\Pi zero 2$. detect that the opposite implication $(\star\star)\Rightarrow(\star)$ holds trivially for any $X$ and $Y$. however the implication $(\star)\Rightarrow (\star\star)$ for an arbitrary Borel set $X \subset 2omega\times2omega$ is reminiscent of the assertion ""$\forall \alpha\in \omegaomega, \,\aleph 1$ is inaccessible in $L(\alpha)$"". extra exactly the authors turn out that the validity of $(\star)\Rightarrow(\star\star)$ for all $X \in \varSigma0 {1 \xi 1 $, is corresponding to ""$\aleph \xi \aleph 1$"". although we will convey independently, that once $X$ is Borel possible, in $ZFC$, derive from $(\star)$ the weaker end that $Y$ is usually Borel and of an analogous Baire classification as $X$. This final consequence solves an previous challenge approximately compact protecting mappings. in truth those effects are heavily with regards to the subsequent common boundedness precept Lift$(X, Y)$: ""If any compact subset of $Y$ admits a continual lifting in $X$, then $Y$ admits a continuing lifting in $X$"", the place through a lifting of $Z\subset \pi(X)$ in $X$ we suggest a mapping on $Z$ whose graph is contained in $X$. the most results of this paintings will provide the precise set theoretical power of this precept looking on the descriptive complexity of $X$ and $Y$. The authors additionally turn out the same outcome for a edition of Lift$(X, Y)$ within which ""continuous liftings"" are changed via ""Borel liftings"", and which solutions a question of H. Friedman. between different purposes the authors receive an entire option to an issue which works again to Lusin about the lifestyles of $\mathbf{\Pi 1 1$ units with all components in a few given category $\mathbf{\Gamma $ of Borel units, bettering prior effects by means of J. Stern and R. Sami. The facts of the most consequence will depend upon a nontrivial illustration of Borel units (in $ZFC$) of a brand new kind, related to a large number of ""abstract algebra"". This illustration was once at the beginning constructed for the needs of this facts, yet has a number of different purposes.

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Borel Liftings of Borel Sets: Some Decidable and Undecidable Statements

One of many goals of this paintings is to enquire a few common houses of Borel units that are undecidable in $ZFC$. The authors' place to begin is the subsequent uncomplicated, notwithstanding non-trivial end result: think about $X \subset 2omega\times2omega$, set $Y=\pi(X)$, the place $\pi$ denotes the canonical projection of $2omega\times2omega$ onto the 1st issue, and think that $(\star)$ : ""Any compact subset of $Y$ is the projection of a few compact subset of $X$"".

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J ' J+l D S j €1 ' S cu . )|2d? )|2d§ . (X'1(x))a(\"1(x)) = xa(X^(x)) = xa(x, j) . 7. ) is constant for some n . If there is point spectrum, then the corresponding eigenspace is infinite dimensional. 7. »-J Chapters 3 and 4 are devoted to the analytic properties of the family of operators case. Lp . In chapter 3 we will show that The analysis is quite simple. parabolic case L- is holomorphic on Le is entire in the elliptic In chapter 4 we will prove that in the (0,2TT) . The singularities at 0 and 2TT are not removable.

2d? )|2d§ . (X'1(x))a(\"1(x)) = xa(X^(x)) = xa(x, j) . 7. ) is constant for some n . If there is point spectrum, then the corresponding eigenspace is infinite dimensional. 7. »-J Chapters 3 and 4 are devoted to the analytic properties of the family of operators case. Lp . In chapter 3 we will show that The analysis is quite simple. parabolic case L- is holomorphic on Le is entire in the elliptic In chapter 4 we will prove that in the (0,2TT) . The singularities at 0 and 2TT are not removable.

In the sequel we will refer to these as classes of type I, II and III respectively. 10) : T h(kT ) 2 2 k=1| kT T/2 i t y T / 2 e e c o s k ? -kTT/2 - i H T / 2 | -e e - 2 ikn? 2 T h(kT ) -e V gT gT r v-i „ kT /2 ikT(j> /2 -kT /2 -ike)) /2 k= 1 r m n •» i n ' n i^ mn mn IgT i gT e gT e _e gT gT e • J2 Observe that the first sum on the right hand side is independent of ? The { T : g € T } appearing in this sum are the lengths of the closed 3 geodesies on E /T . Each such length is represented by infinitely many 3 distinct closed geodesies on E fT .

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