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\(\sigma\)-collectionwise Hausdorffness at singular strong limit cardinals. (English) Zbl 1107.54023

A topological space \(Y\) is called separated if there is a function \(y\mapsto U_y\) such that for all \(y,y'\in Y\), \(U_y\) is open, \(y\in U_y\), and if \(U_y\neq U_{y'}\), then \(U_y\cap U_{y'} = \emptyset\). A space \(X\) is called \(\kappa\)-collectionwise Hausdorff (\(\kappa\)-\(\sigma\)-collectionwise Hausdorff) if every closed discrete subspace \(Y\subseteq X\) is separated (a countable union of separated sets). For a fixed cardinal \(\kappa\), consider the statements
CWH\(_\kappa\): If \(X\) is a normal topological space and there is some \(\lambda <\kappa\) such that every point in \(X\) has a neighbourhood base of size \(\leq\lambda\), then \(X\) is \(\kappa\)-collectionwise Hausdorff, and
\(\sigma\)CWH\(_\kappa\): If \(X\) is a \(\nearrow\)-normal (for a definition, see p.1501 of the paper under review) topological space and there is some \(\lambda <\kappa\) such that every point in \(X\) has a neighbourhood base of size \(\leq\lambda\), then \(X\) is \(\kappa\)-\(\sigma\)-collectionwise Hausdorff.
In [W. G. Fleissner, Set-theor. Topol., Vol. dedic. to M. K. Moore, 135–140 (1977; Zbl 0375.54005) and Z. T. Balogh and D. K. Burke, Topology Appl. 57, No. 1, 71–85 (1994; Zbl 0849.54013)], the authors have proved that under the assumption GCH, the properties CWH\(_\kappa\) and \(\sigma\)CWH\(_\kappa\) hold for all singular cardinals \(\kappa\) (Fleissner for CWH\(_\kappa\) and Balogh and Burke for \(\sigma\)CWH\(_\kappa\)). In [Fundam. Math. 138, No. 1, 59–67 (1991; Zbl 0766.54016)], the present author had proved Fleissner’s result under the weaker assumption SCH (the singular cardinals hypothesis). In the paper under review, he proves the Balogh-Burke result under the same hypothesis.

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
03E10 Ordinal and cardinal numbers
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References:

[1] Balogh, Z. T.; Burke, D. K., On ↗-normal spaces, Topology Appl., 57, 71-85 (1994) · Zbl 0849.54013
[2] Fleissner, W. G., Separating closed discrete collections of singular cardinality, (Reed, G. M., Set Theoretic Topology (1977), Academic Press: Academic Press New York), 135-140 · Zbl 0375.54005
[3] Jech, T., Set Theory (1978), Academic Press: Academic Press New York · Zbl 0419.03028
[4] Kemoto, N., Collectionwise Hausdorffness at limit cardinals, Fund. Math., 138, 59-67 (1991) · Zbl 0766.54016
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