Reflection theorems for cardinal functions. (English) Zbl 0943.54003

A reflection theorem is a statement of the form “every structure with a certain property has a small substructure with the same property”. Normally ‘smaller’ means ‘of cardinality \(\kappa\) or less’ for certain \(\kappa\). In this paper a cardinal function \(\varphi\) is said to (strongly) reflect \(\kappa\) if every space \(X\) with \(\varphi(X)\geq\kappa\) has a subspace \(Y\) with \(|Y|\leq\kappa\) and \(\varphi(Y)\geq\kappa\) (\(\varphi(Z)\geq\kappa\) whenever \(Y\subseteq Z\subseteq X\)). A model result is A. Hajnal and I. Juhász’s theorem [Proc. Am. Math. Soc. 79, 657-658 (1980; Zbl 0432.54003)] that weight strongly reflects every infinite cardinal. The authors prove reflection theorems for many familiar cardinal functions, e.g., \(c\), \(s\), \(e\), \(hd\) and \(hL\) strongly reflect all infinite cardinals; and \(t\) and \(\chi\) reflect all cardinals in the class of compact Hausdorff spaces. The final section contains a generalization of A. Dow’s metrization theorem [ibid. 104, No. 3, 999-1001 (1988; Zbl 0692.54018)] to higher cardinals: if \(X\) is initially \(\kappa\)-compact, \(\chi(X)\leq\kappa\) and \(w(X)\geq\kappa^+\) then there is \(Y\subseteq X\) of cardinality at most \(\kappa^+\) such that \(pw(Y)\geq\kappa^+\), where \(pw\) is the point-weight, the minimum order of a base for the space.
Reviewer: K.P.Hart (Delft)


54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E10 Ordinal and cardinal numbers
54A35 Consistency and independence results in general topology
54D30 Compactness
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