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A cardinal generalization of z-embedding. (English) Zbl 0572.54011
Rings of continuous functions, Pap. Spec. Sess. Annu. Meet. Am. Math. Soc., Cincinatti/Ohio 1982, Lect. Notes Pure Appl. Math. 95, 7-66 (1985).
[For the entire collection see Zbl 0553.00006.]
From the author’s introduction: ”It is known, on the one hand, that $$P^{\kappa}$$-embedding may be viewed as a cardinal generalization of C- embedding (in fact, by results of H. L. Shapiro [Can. J. Math. 18, 981-998 (1966; Zbl 0158.413)] and T. E. Gantner [Trans. Am. Math. Soc. 132, 147-157 (1968; Zbl 0157.535)] $$P^{\omega}$$-embedding$$=C$$- embedding), and, on the other, that C-embedding$$=z$$-embedding$$+well$$- embedding, in our paper with A. W. Hager [Math. Z. 136, 41-52 (1974; Zbl 0264.54011)]. In this paper we cardinally generalize the notion of z-embedding to that of $$''z_{\kappa}$$-embedding” in such a way that (1) $$z_{\omega}$$-embedding$$=z$$-embedding, (2) $$z_{\kappa}$$- embedding relates to $$P^{\kappa}$$-embedding in precisely the same way that z-embedding relates to C-embedding and (3) virtually all of the basic results concerning z-embedding generalize satisfactorily to the case of $$z_{\kappa}$$-embedding. Section 2 is devoted to some technical preliminaries, including uniform discreteness (a main tool of this paper), normal covers, and mappings into hedgehogs.
In Section 3 we define $$z_{\kappa}$$-embedding and establish some of its basic properties.
Interconnections between $$z_{\kappa}$$-embedding, well-embedding, and $$P^{\kappa}$$-embedding are studied in Section 4. We newly characterize well-embedding and $$P^{\kappa}$$-embedding, respectively, and apply the techniques to obtain completely new proofs of Przymusiński’s and Morita’s characterizations of $$P^{\kappa}$$-embedding (in terms of extendibility of maps into hedgehogs [see T. Przymusiński, Fundam. Math. 98, 75-81 (1978; Zbl 0382.54014)] or into complete absolute retracts for metrizable spaces [see K. Morita, ibid. 88, 1-6 (1975; Zbl 0304.55009)].
Section 5 deals with $$P^{\kappa}$$-embedding of the Tychonoff space X in its $$\alpha$$-compactification $$\beta_{\alpha}X$$ [in the sense of H. Herrlich, Math. Z. 96, 64-72 (1967; Zbl 0149.195), where $$\alpha$$ is an infinite cardinal].
The final three sections are concerned with spaces in which special subsets are $$z_{\kappa}$$-embedded. In Sections 6 and 7, for example, we show (1) that a space X is $$\kappa$$-collectionwise normal if and only if every nowhere dense closed subset of X is $$z_{\kappa}$$-embedded in X [which generalizes results of C. H. Dowker in Ark. Mat. 2, 307-313 (1952; Zbl 0048.410), and H. L. Shapiro (loc. cit.) and Fundam. Math. 66, 263-281 (1970)], (2) that every normally placed subset of a $$\kappa$$-collectionwise normal space X is $$z_{\kappa}$$-embedded in X and (3) that every subset of a perfectly $$\kappa$$-collectionwise normal space X is $$z_{\kappa}$$-embedded in X.
And finally, in Section 8, we study the class $$Oz_{\kappa}$$ of those spaces X in which every open subset of X is $$z_{\kappa}$$-embedded. $$\{$$ The class Oz $$(=Oz_{\omega})$$ was introduced by the author [Can. J. Math. 28, 673-690 (1976; Zbl 0359.54009)], and independently by E. V. Shchepin [Sov. Math., Dokl. 17, 152-155 (1976; Zbl 0338.54022)] and by E. P. Lane [Pac. J. Math. 82, 155-162 (1970; Zbl 0386.54006)] (with different terminology) and by T. Terada [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 13, 129-132 (1975; Zbl 0333.54008)]$$\}$$. Most of the known characterizations of Oz are generalized to $$Oz_{\kappa}$$. We show that $$\kappa <m(\alpha)$$ $$(=$$ the smallest measurable cardinal $$\geq \alpha)$$ if and only if, for every Tychonoff space X, X is $$Oz_{\kappa}$$ (if and) only if $$\beta_{\alpha}X$$ is $$Oz_{\kappa}.''$$
Reviewer: C.E.Aull

##### MSC:
 54C45 $$C$$- and $$C^*$$-embedding 54C50 Topology of special sets defined by functions