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On closed mappings. (English) Zbl 0665.54008

If a space X is homeomorphic with a subspace of Y then X has smaller or equal dimensional type than Y, denoted by dX\(\leq dY\). The author generalizes dimension type as follows: If X is the image of a subspace of Y under a closed map, then X has smaller or equal closed-map type, than Y, denoted by ctX\(\leq ctY\). If ctX\(\leq ctY\) and it is not true that ctY\(\leq ctX\), then X has smaller closed-type than Y, i.e. \(ctX<ctY\). If neither ctX\(\leq ctY\) or ctY\(\leq ctX\) holds, then X and Y are incomparable.
The author generalizes a theorem of K. Kuratowski [Fundam. Math. 8, 201-208 (1926)] on dimension type. Theorem 2. In any hereditarily separable, hereditarily normal and first countable space of cardinality of the continuum there is a family of \(2^ c\) subspaces whose closed-map types are incomparable. The author generalizes a Theorem of S. Banach [Fundam. Math. 19, 10-16 (1932; Zbl 0005.19602)] as follows. Theorem 3. If X is an unperfect, hereditarily separable, hereditarily paracompact and first countable space with \(| X| =c\) and Y is a regular and first countable space with \(| Y| \leq c\), then there exists a family \(\{A_{\alpha}:\) \(\alpha <c\}\) such that: (i) for each \(\alpha <c\), there is an \(A_{\alpha}\subset X\) such that \(| A|_{\alpha}=c\), (ii) for each \(B\subset Y\), if there are distinct \(\alpha,\beta <c\), with \(ctB\leq ctA_{\alpha}\) and \(ctB\leq ctA_{\beta}\), then \(| B| <c.\)
The remaining portion of the paper deals with separable metric spaces. Some sample theorems are: If X is separable metric and unperfect, then \(ctX<ctC\), where C is the Cantor set. If X is separable metric and \(ctX<ctC\), then there exists a metric space Y with \(ctX<ctY<ctC\).
Reviewer: E.Duda

MSC:

54C10 Special maps on topological spaces (open, closed, perfect, etc.)

Citations:

Zbl 0005.19602
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References:

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