## 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)

### Citations:

Zbl 0134.18801; Zbl 0872.54002
Full Text:

### References:

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