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A classification of separable Rosenthal compacta and its applications. (English) Zbl 1146.54007

The structure theory of separable Rosenthal compacta (i.e., compact subspaces of the Baire class one functions on a Polish space with the topology of pointwise convergence) is developed. The main result of the paper delineates seven separable Rosenthal compact spaces \((K_i:i<7)\), and asserts that any other separable Rosenthal compact space includes one of the seven in a canonical way. More precisely, if \(X\) and \(Y\) are Polish and \(F=\{f_t:t\in 2^{<{\mathbb N}}\}\subseteq {\mathcal B}_1(X)\) and \(G=\{g_t:t\in 2^{<{\mathbb N}}\}\subseteq {\mathcal B}_1(Y)\) are relatively compact, then \(F\) and \(G\) are said to be equivalent if the natural mapping \(f_t\to g_t\) extends to a homeomorphism of \(\overline F\) and \(\overline G\) (with respect to the pointwise topology). A family \(F=\{f_t:t\in 2^{<{\mathbb N}}\}\subseteq {\mathcal B}_1(X)\) is said to be minimal if for any dyadic subtree \(S\subseteq 2^{<{\mathbb N}}\), \(\{f_t:t\in 2^{<{\mathbb N}}\}\) is equivalent to \(\{f_s:s\in S\}\) with respect to the natural isomorphism between \(S\) and \(2^{<{\mathbb N}}\).
The main result asserts that there are seven minimal families of functions of the form \(\{f_t:t\in 2^{<{\mathbb N}}\}\subseteq {\mathcal B}_1({\mathbb R})\) such that each of the corresponding pointwise closures are Rosenthal and contain the minimal family as a discrete subspace. Moreover, given any separable Rosenthal compacta \(K\) containing a dense set \(\{f_n:n\in \omega\}\) one of the seven minimal families canonical embeds into \(K\) in the sense that there is an increasing map \(\varphi:2^{<{\mathbb N}}\to {\mathbb N}\) such that \(\{f_{\varphi(t)}:t\in 2^{<{\mathbb N}}\}\) is equivalent to one of the seven minimal families. Many of the results are natural refinements of previous work in the area, e.g., Todorcevic’s result that the split interval (double arrow space) embeds into any non-metrizable, hereditarily separable Rosenthal compact, R. Pol’s result that every non-hereditarily separable Rosenthal compacta contains an uncountable discrete subspace of size \(2^\omega\), and others.
Similar structural results are proven for “analytic” subspaces of a separable Rosenthal compacta \(K\): A subspace \(C\subseteq K\) is said to be analytic with respect to a countable dense subset \(\{f_n:n\in \omega\}\) of \(K\) if there is an analytic set \(A\subseteq [\mathbb N]^{\aleph_0}\) such that the accumulation points of \(\{f_n:n\in L\}\) lie in \(C\) whenever \(L\in A\), and every \(g\in C\) is a limit point of \(\{f_n:n\in L\}\) for some \(L\in A\). These structural results for analytic subspaces are then applied to the study of the existence of unconditional families in Banach spaces. The most quotable result is that if \(X\) is a nonseparable representable Banach space, then \(X^*\) contains an unconditional family of size \(| X^*| \).

MSC:

54C35 Function spaces in general topology
03E15 Descriptive set theory
05C05 Trees
05D10 Ramsey theory
26A21 Classification of real functions; Baire classification of sets and functions
46B03 Isomorphic theory (including renorming) of Banach spaces
46B26 Nonseparable Banach spaces
54D30 Compactness
54D55 Sequential spaces
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