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

By Gabriel Debs

One of many goals of this paintings is to enquire a few average homes of Borel units that are undecidable in $ZFC$. The authors' place to begin is the next uncomplicated, even though non-trivial end result: give some thought to $X \subset 2omega\times2omega$, 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 furthermore $X$ is $\mathbf{\Pi zero 2$ then $(\star\star)$: ""The limit of $\pi$ to a couple really closed subset of $X$ is ideal onto $Y$"" it follows that during the current case $Y$ can also be $\mathbf{\Pi zero 2$. 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 comparable to the assertion ""$\forall \alpha\in \omegaomega, \,\aleph 1$ is inaccessible in $L(\alpha)$"". extra exactly the authors end up that the validity of $(\star)\Rightarrow(\star\star)$ for all $X \in \varSigma0 {1 \xi 1$, is comparable to ""$\aleph \xi \aleph 1$"". although we will express independently, that once $X$ is Borel possible, in $ZFC$, derive from $(\star)$ the weaker end that $Y$ is usually Borel and of an identical Baire type as $X$. This final end result solves an previous challenge approximately compact overlaying mappings. in truth those effects are heavily on the topic of the next normal boundedness precept Lift$(X, Y)$: ""If any compact subset of $Y$ admits a continual lifting in $X$, then $Y$ admits a continual lifting in $X$"", the place via 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 supply the precise set theoretical power of this precept reckoning on the descriptive complexity of $X$ and $Y$. The authors additionally turn out the same consequence for a version of Lift$(X, Y)$ within which ""continuous liftings"" are changed through ""Borel liftings"", and which solutions a query of H. Friedman. between different purposes the authors receive an entire technique to an issue which fits again to Lusin about the lifestyles of $\mathbf{\Pi 1 1$ units with all ingredients in a few given type $\mathbf{\Gamma$ of Borel units, bettering previous effects via J. Stern and R. Sami. The evidence of the most end result will depend on a nontrivial illustration of Borel units (in $ZFC$) of a brand new variety, regarding a large number of ""abstract algebra"". This illustration was once before everything built for the needs of this evidence, yet has numerous different functions.

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L1 o(E). l1 o(E) consider the decreasing sequence of the dimensions of the vector k-spaces L,/uLo, where L, is the A-module of the x E E such that [q'ex] (x) « 1, and t varies from to 1 ; we call this sequence, the sequence of invariants of ex. l1 o(E), it is necessary and sufficient that Xo = X~ (exerc. 12, e» and that the sequence of the invariants of ex and of ~ should be the same (use exerc. 12, b». l1 o(E)/GL(E) is isomorphic with the space of the orbits T"/G n , where the symmetric group operates on the right on T" by ° ° (ZI' ...

COROLLARY 4. Convex cones 3. -A subset C ofan affine space E is a cone with vertex Xo if C is invariant for all homotheties of centre Xo and ratio > o. We shall suppose in this No. e. we suppose that E is a vector space, and when we speak of a cone, it is to be understood that this cone has vertex O. The set of points of the form Aa for A > 0 (resp. A ~ 0), where a is a non-null vector, is called an open half line (resp. closed half-line) originating at O. A cone C of vertex 0 is said to be pointed if 0 E C, and non-pointed otherwise.

The closed segment with end points x, y is contained in C; ifit contains 0 then AX + (1 - A) y = 0 for some Awith 0 < A < 1, therefore x = IlY with 11 < O. Thus C contains the line through 0 and x, contrary to hypothesis. PROPOSITION 10. f C + C c C and AC c C for all A > O. For the condition AC c C for all A > 0 characterises the cones. If C is convex we have C + C = + = C (II, p. 8, Remark). Conversely, if the cone C is such thatC + C c C, then for 0 < A < 1, wehaveAC + (1 - A) C = C + C c C, which shows that C is convex.