Vitalij A. Chatyrko*
Linköping University, Sweden
Counterparts of Smirnov's compacta for inductive functions PtrInd
In 1959 Smirnov constructed metrizable compacta $S^\alpha, \ \alpha < \omega_1$, such that $trInd S^\alpha = \alpha$ for each $\alpha$. Some years later Levshenko proved that $trInd\,X \leq \omega_0 \cdot trind\,X$ for any metrizable compact space $X$.
This result together with the inductive character of the function $trind$ implied for each $\alpha: 0 \leq \alpha < \omega_1,$ the existence of
a compact metrizable space $X_\alpha$ such that
$trind\,X_\alpha = \alpha \leq trInd\,X_\alpha \ne \infty$.
We generalize Smirnov's construction.
In particular, for each absolute multiplicative or additive Borel class $P$ and each $\alpha < \omega_1$ we present a separable metrizable space $S^\alpha_{P}$ such that $PtrInd\,S^\alpha_{P} = trInd\,S^\alpha_{P} = \alpha$ and $QtrInd\,S^\alpha_{P} = -1$ for any other absolute multiplicative or additive Borel class $Q$ containing $P$.
In 1997 Charalambous proved that for any separable metrizable space $X$ with $trInd\,X \ne \infty$ and any absolute multiplicative or additive Borel class $P$ the inequality $PtrInd\,X \leq \omega_0 \cdot (Ptrind\,X +1)$ holds.
These two results imply that for each absolute multiplicative or additive Borel class $P$ and each $\alpha < \omega_1$ there exists a separable metrizable space $X^\alpha_{P}$ such that $Ptrind\,X^\alpha_{P} = \alpha \leq trInd\,X^\alpha_{P} \ne \infty$ and $QtrInd\,S^\alpha_{P} = -1$ for any other absolute multiplicative or additive Borel class $Q$ containing $P$.
* This is a joint work with Yasunao Hattori.
