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A factorization theorem for the transfinite kernel dimension of metrizable spaces. (English) Zbl 0905.54022

For a space \(X\) one defines \(D_\alpha(X)\) and \(E_\alpha(X)\) inductively by \(D_{-1}(X)=\emptyset\), \(E_\alpha(X)=X\smallsetminus \bigcup_{\beta<\alpha}D_\beta(X)\) and \(D_{\lambda+n}(X)=\bigcup\{U:U\) is open in \(E_\lambda(X)\) and \(\dim U\leq n\}\), here \(\lambda\) denotes a limit ordinal and \(n\) a finite ordinal. The transfinite kernel dimension of \(X\), denoted \(\text{trker }X\), is the first ordinal \(\alpha\) for which \(D_\alpha(X)=X\), if any; it is \(\infty\) otherwise.
The author proves a factorization theorem for this dimension function in the class of metrizable spaces and in the spirit of B. A. Pasynkov’s theorem [ibid. 60, 285-308 (1967; Zbl 0168.43801)]: if \(f:X\to Y\) is continuous then there are a space \(Z\), with \(\text{trker }Z\leq \text{trker }X\) and \(w(Z)\leq\max\{| \text{trker }X| ,w(Y)\}\), and continuous maps \(g:X\to Z\) and \(h:Z\to Y\) such that \(f=h\circ g\). One may then apply Pasynkov’s general method [loc. cit.] to show that for every ordinal \(\mu\) and every cardinal \(\kappa\) there is a universal space in the class of metrizable spaces of weight at most \(\kappa\) and transfinite kernel dimension at most \(\mu\) – a result obtained by W. Olszewski [ibid. 140, No. 1, 35-48 (1991; Zbl 0807.54007)].
Reviewer: K.P.Hart (Delft)

MSC:

54F45 Dimension theory in general topology
54E35 Metric spaces, metrizability
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