Reflection theorems for some cardinal functions. (English) Zbl 1144.54003

The reflection theorems in the title are in the spirit of A. Hajnal and I. Juhász [Proc. Am. Math. Soc. 79, 657–658 (1980; Zbl 0432.54003)] and R. E. Hodel and J. E. Vaughan [Topology Appl. 100, No. 1, 47–66 (2000; Zbl 0943.54003)]: if \(\varphi\) is a cardinal function and \(\kappa\) a cardinal then ‘\(\varphi\) reflects \(\kappa\)’ means that every space \(X\) with \(\varphi(X)\geq\kappa\) has a subspace \(Y\) such that \(| Y| \leq\kappa\) and \(\varphi(Y)\geq\kappa\). Thus the main result of A. Hajnal and I. Juhász [loc. cit.] states that weight reflects every cardinal.
Aided by convenient formulas for the weight the authors show reflection of \(\kappa\) for density, the (weak) Lindelöf number, the boundedness index in the class of topological groups of character less that \(\kappa\). Dually, if the other functions are kept in check then character reflects \(\kappa\) (or \(\kappa^+\)). The authors also prove reflection results for more exotic cardinal functions that are inspired by Šapirovskiĭ’s lemma [B. Šapirovskiĭ, Sov. Math. Dokl. 13, 215–219 (1972; Zbl 0252.54002)] that given an open cover \(\mathcal{U}\) one can find \(A\) and \(\mathcal{U}'\subseteq\mathcal{U}\) such that, \(A\) is discrete, \(X=\overline{A}\cup\bigcup\mathcal{U}'\) and \(| A| =| \mathcal{U}'| \). These functions prescribe the sizes of \(A\) and \(\mathcal{U}'\), and/or drop the closure condition.
Reviewer: K. P. Hart (Delft)


54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)