## 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)

### MSC:

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

### Keywords:

reflection theorem; cardinal functions

### Citations:

Zbl 0432.54003; Zbl 0943.54003; Zbl 0252.54002