Codings of separable compact subsets of the first Baire class - Mathematics > Logic

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Abstract: Let $X$ be a Polish space and $K$ a separable compact subset of the firstBaire class on $X$. For every sequence $\bs$ dense in $\kk$, the descriptiveset-theoretic properties of the set \ \lbf=\{L\in n: f n {n\in L}\text{is pointwise convergent}\} \ are analyzed. It is shown that if $K$ isnot first countable, then $\lbf$ is $\PB^1 1$-complete. This can also happeneven if $K$ is a pre-metric compactum of degree at most two, in the sense of S.Todorcevic. However, if $K$ is of degree exactly two, then $\lbf$ is alwaysBorel. A deep result of G. Debs implies that $\lbf$ contains a Borel cofinalset and this gives a tree-representation of $\kk$. We show that classicalordinal assignments of Baire-1 functions are actually $\PB^1 1$-ranks on $\kk$.We also provide an example of a $\SB^1 1$ Ramsey-null subset $A$ of $n$ forwhich there does not exist a Borel set $B\supseteq A$ such that the difference$B\setminus A$ is Ramsey-null.

Author: Pandelis Dodos

Source: https://arxiv.org/