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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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