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Some factorization theorems for paracompact \(\sigma\)-spaces. (English) Zbl 0636.54031
Factorization theorems in dimension theory are proved. A continuous map \(f: X\to Y\) of a regular space X onto a paracompact space Y is said to be \(\sigma\)-discrete if there exists a \(\sigma\)-discrete network \({\mathcal K}\) in X such that f(\({\mathcal K})\) is a \(\sigma\)-discrete network in Y. This concept stems from a weak bijection due to A. V. Arkhangel’skij [Usp. Mat. Nauk 36, No.3(219), 127-146 (1981; Zbl 0468.22001)]. An inverse system \(\{X_{\alpha},{\mathcal F}_{\alpha},\pi^{\alpha}_{\beta}\), \(\alpha\),\(\beta\in A\}\) is said to be quasi-rigid if all \(X_{\alpha}\) are paracompact spaces with a \(\sigma\)-discrete network \({\mathcal F}_{\alpha}\) and \(\pi^{\alpha}_{\beta}({\mathcal F}_{\alpha})={\mathcal F}_{\beta}\) for each ordered pair \(\beta\leq \alpha\). A quasi-rigid system is said to be rigid if all \(\pi^{\alpha}_{\beta}\) are continuous bijections. (1) X is a paracompact \(\sigma\)-space with dim \(X\leq n\) if it is homeomorphic to a limit of a quasi-rigid system of metric spaces \(X_{\alpha}\) with dim \(X_{\alpha}\leq n\). (2) For every \(\sigma\)-discrete map \(f: X\to Y\) there exist a paracompact \(\sigma\)-space Z and \(\sigma\)-discrete maps \(g: X\to Z\) and \(h: Z\to Y\) such that dim \(Z\leq \dim X\), w(Z)\(\leq w(Y)\) and \(f=h\circ g\). (3) For every closed map \(f: X\to Y\) of a paracompact \(\sigma\)-space X there exist a paracompact \(\sigma\)-space Z and continuous maps \(g: X\to Z\) and \(h: Z\to Y\) such that dim \(Z\leq \dim X\), w(Z)\(\leq w(Y)\) and \(f=h\circ g\). (4) For every continuous map \(f: X\to Y\) of a regular space X with a countable network onto a regular space Y there exist a regular space Z with a countable network and a continuous maps \(g: X\to Z\) and \(h: Z\to Y\) such that dim \(Z\leq \dim X\), w(Z)\(\leq w(Y)\) and \(f=h\circ g\).
Reviewer: K.Nagami

MSC:
54F45 Dimension theory in general topology
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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