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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)

##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 03E10 Ordinal and cardinal numbers 54A35 Consistency and independence results in general topology 54D30 Compactness
##### Keywords:
cardinal function; compactness; weight; point-weight
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