General embedding properties of absolute Borel and Souslin spaces

Hansell, Roger W.

doi:10.1090/S0002-9939-1971-0268853-X

General embedding properties of absolute Borel and Souslin spaces
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by Roger W. Hansell PDF

Proc. Amer. Math. Soc. 27 (1971), 343-352 Request permission

Abstract:

In a recent paper, S. Willard established several characterizations of absolute metric ${G_\alpha }$-spaces in terms of the Borel character they possessed as subspaces of certain compact Hausdorff spaces; and he asks whether a similar result holds for the ${F_\alpha }$-spaces. In the present paper, we show that for a metric space $X$ the following are equivalent for $\alpha \geqq 2$ : (1) $X$ is an absolute metric ${F_\alpha }$-space, (2) $X$ is a ${Z_\alpha } \bigcap {G_\delta }$-set (i.e., a Baire set of class $\alpha$ intersected with a ${G_\delta }$-set) in some compactification, (3) $X$ is an ${F_\alpha } \bigcap {G_\delta }$-set in every completely regular Hausdorff embedding, (4) $X$ is an absolute ${F_\alpha }$-space with respect to the class of all perfectly normal spaces. These properties remain equivalent when “${F_\alpha }$” and “${Z_\alpha }$” are replaced by “Souslin.” Necessary and sufficient conditions for a metric space to be an ${F_\alpha }$-set in all its compactifications are found and, throughout, extensions to spaces which are not necessarily metrisable are provided.

References

Gustave Choquet, Ensembles ${\cal K}$-analytiques et ${\cal K}$-sousliniens. Cas général et cas métrique, Ann. Inst. Fourier (Grenoble) 9 (1959), 75–81 (French). MR 112843
R. Engelking, Outline of general topology, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. Translated from the Polish by K. Sieklucki. MR 0230273
Z. Frolík, A note on $C(P)$ and Baire sets in compact and metrizable spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 779–784 (English, with Russian summary). MR 224480
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
Felix Hausdorff, Set theory, 2nd ed., Chelsea Publishing Co., New York, 1962. Translated from the German by John R. Aumann et al. MR 0141601

Topologie

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Stelios Negrepontis, Absolute Baire sets, Proc. Amer. Math. Soc. 18 (1967), 691–694. MR 214031, DOI 10.1090/S0002-9939-1967-0214031-9
A. H. Stone, Absolute $F_{\sigma }$ spaces, Proc. Amer. Math. Soc. 13 (1962), 495–499. MR 138088, DOI 10.1090/S0002-9939-1962-0138088-4
S. Willard, Absolute Borel sets in their Stone-Čech compactifications, Fund. Math. 58 (1966), 323–333. MR 196706, DOI 10.4064/fm-58-3-323-333
Stephen Willard, Embedding metric absolute Borel sets in completely regular spaces, Colloq. Math. 20 (1969), 83–88. MR 242124, DOI 10.4064/cm-20-1-83-88