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Transfinite large inductive dimensions modulo absolute Borel classes. (English) Zbl 1173.54014

The notion of small or large inductive dimension modulo a class \(\mathcal P\) of topological spaces was introduced by A. Lelek [Colloq. Math. 12, 221–227 (1964; Zbl 0134.18801)]. To produce such a small relative dimension, start by saying that a space \(X\) has \(\mathcal P\)-ind\(X=-1\) if \(X\in\mathcal P\); then use the inductive notions for defining the higher small inductive dimensions in accordance say with the techniques that might be found in [R. Engelking, Theory of dimensions, finite and infinite. Lemgo: Heldermann (1995; Zbl 0872.54002)] for small inductive dimension which begin with ind\(X=-1\) if and only if \(X=\emptyset\). The same could be done for large inductive dimension and for small and large transfinite inductive dimensions.
In this paper the spaces \(X\) are separable and metrizable. The classes \(\mathcal P\) will be taken from among the respective absolutely additive and absolutely multiplicative Borel classes, \(A(\alpha)\), \(B(\alpha)\), \(0\leq\alpha<\omega_1\).
There are two problems in play. For Problem 1.1, let d be either ind or Ind, and consider two functions, \(a\), \(m:[0,\omega_1)\to\mathbb Z_{\geq -1}\cup\{\infty\}\). They are required to satisfy,
(i)
\(a(0)\geq m(0)\geq\max\{a(1),m(1)\}\), and
(ii)
\(\min\{a(\alpha),m(\alpha)\}\geq\max\{a(\beta),m(\beta)\}\), if \(1\leq\alpha<\beta<\omega_1\).
Does there exist a space \(X\) such that \(A(\alpha)\)-d\(X=a(\alpha)\) and \(M(\alpha)\)-d\(X=m(\alpha)\) for each \(0\leq\alpha<\omega_1\)? This problem is solved in the affirmative in the case that \(d=\)Ind (Corollary 4.2).
For Problem 1.2, let d be either trind or trInd, and consider two functions, \(a\), \(m:[0,\omega_1)\to[-1,\Omega)\cup\{\infty\}\) where \(\Omega\) is the first uncountable ordinal. They are required to satisfy,
(i)
\(a(0)\geq m(0)\geq\max\{a(1),m(1)\}\), and
(ii)
\(\min\{a(\alpha),m(\alpha)\}\geq\max\{a(\beta),m(\beta)\}\), if \(1\leq\alpha<\beta<\omega_1\).
Does there exist a space \(X\) such that \(A(\alpha)\)-d\(X=a(\alpha)\) and \(M(\alpha)\)-d\(X=m(\alpha)\) for each \(0\leq\alpha<\omega_1\)? This one is solved also in the affirmative (Theorem 4.1).

MSC:

54F45 Dimension theory in general topology
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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References:

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