论文信息 - The Dyck and the Preiss separation uniformly

The Dyck and the Preiss separation uniformly

Abstract We are concerned with two separation theorems about analytic sets by Dyck and Preiss, the former involves the positively-defined subsets of the Cantor space and the latter the Borel-convex subsets of finite dimensional Banach spaces. We show by introducing the corresponding separation trees that both of these results admit a constructive proof. This enables us to give the uniform version of these separation theorems, and to derive as corollaries the results, which are analogous to the fundamental fact “ HYP is effectively bi-analytic” provided by the Suslin–Kleene Theorem.

Vassilios Gregoriades

[1] L. Ros. On the Structure of Finite Level and ω-Decomposable Borel Functions , 2012, The Journal of Symbolic Logic.

[2] Arno Pauly,et al. Non-deterministic computation and the Jayne-Rogers Theorem , 2012, DCM.

[3] Clifford Spector,et al. Recursive well-orderings , 1955, Journal of Symbolic Logic.

[4] Martin Ziegler,et al. Computable operators on regular sets , 2004, Math. Log. Q..

[5] Jr. V. L. Klee. Convex sets in linear spaces. III. , 1951 .

[6] The convex generation of convex Borel sets in locally convex spaces , 1974 .

[7] Vassilios Gregoriades. Turning Borel sets into clopen sets effectively , 2012 .

[8] Alain Louveau,et al. A separation theorem for Σ¹₁ sets , 1980 .

[9] Arno Pauly,et al. Finite choice, convex choice and finding roots , 2015 .

[10] Alain Louveau,et al. A Separation Theorem for ∑ 1 1 Sets , 1980 .

[11] Y. Moschovakis. Descriptive Set Theory , 1980 .

[12] Takayuki Kihara,et al. Decomposing Borel functions using the Shore-Slaman join theorem , 2013, 1304.0698.

[13] A. Kechris. Classical descriptive set theory , 1987 .

[14] David Preiss. The convex generation of convex Borel sets in Banach spaces , 1973 .