Homogeneous Borel sets of ambiguous class two
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- by Fons van Engelen PDF
- Trans. Amer. Math. Soc. 290 (1985), 1-39 Request permission
Abstract:
We describe and characterize all homogeneous subsets of the Cantor set which are both an ${F_{\sigma \delta }}$ and a ${G_{\delta \sigma }}$; it turns out that there are ${\omega _1}$ such spaces.References
- Paul Alexandroff, Über nulldimensionale Punktmengen, Math. Ann. 98 (1928), no. 1, 89–106 (German). MR 1512393, DOI 10.1007/BF01451582
- R. D. Anderson, On topological infinite deficiency, Michigan Math. J. 14 (1967), 365–383. MR 214041 L. E. J. Brouwer, On the structure of perfect sets of points, Proc. Akad. Amsterdam 12 (1910), 785-794. E. van Douwen, unpublished. F. van Engelen, unpublished.
- Fons van Engelen and Jan van Mill, Borel sets in compact spaces: some Hurewicz type theorems, Fund. Math. 124 (1984), no. 3, 271–286. MR 774518, DOI 10.4064/fm-124-3-270-286
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- R. Engelking, W. Holsztyński, and R. Sikorski, Some examples of Borel sets, Colloq. Math. 15 (1966), 271–274. MR 201314, DOI 10.4064/cm-15-2-271-274
- R. C. Freiwald, R. McDowell, and E. F. McHugh Jr., Borel sets of exact class, Colloq. Math. 41 (1979), no. 2, 187–191. MR 591922, DOI 10.4064/cm-41-2-187-191
- A. Gutek, On extending homeomorphisms on the Cantor set, Topological structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978) Math. Centre Tracts, vol. 115, Math. Centrum, Amsterdam, 1979, pp. 105–116. MR 565830 F. Hausdorff, Grundzüge der Mengenlehre, Leipzig, 1914. —, Die schlichten stetigen Bilder des Nullraums, Fund. Math. 29 (1937), 151-158. W. Hurewicz, Relativ perfekte Teile von Punktmengen und Mengen $(A)$, Fund. Math. 12 (1928), 78-109.
- B. Knaster and M. Reichbach, Notion d’homogénéité et prolongements des homéomorphies, Fund. Math. 40 (1953), 180–193 (French). MR 61817, DOI 10.4064/fm-40-1-180-193 K. Kuratowski, Topologie. I, 2nd ed., PWN, Warsaw, 1948.
- Jan van Mill, Characterization of some zero-dimensional separable metric spaces, Trans. Amer. Math. Soc. 264 (1981), no. 1, 205–215. MR 597877, DOI 10.1090/S0002-9947-1981-0597877-9
- Jan van Mill, Characterization of a certain subset of the Cantor set, Fund. Math. 118 (1983), no. 2, 81–91. MR 732656, DOI 10.4064/fm-118-2-81-91
- A. V. Ostrovskiĭ, Continuous images of the Cantor product $C\times \textbf {Q}$ of a perfect set $C$ and the rational numbers $\textbf {Q}$, Seminar on General Topology, Moskov. Gos. Univ., Moscow, 1981, pp. 78–85 (Russian). MR 656951
- Jean Pollard, On extending homeomorphisms on zero-dimensional spaces, Fund. Math. 67 (1970), 39–48. MR 270348, DOI 10.4064/fm-67-1-39-48
- Doron Ravdin, On extensions of homeomorphisms to homeomorphisms, Pacific J. Math. 37 (1971), 481–495. MR 320992
- Jean Saint-Raymond, La structure borélienne d’Effros est-elle standard?, Fund. Math. 100 (1978), no. 3, 201–210 (French). MR 509546, DOI 10.4064/fm-100-3-201-210 W. Sierpiński, Sur une propriété topologique des ensembles dénombrables denses en soi, Fund. Math. 1 (1920), 11-16. —, Sur une définition topologique des ensembles ${F_{\sigma \delta }}$, Fund. Math. 6 (1924), 24-29.
- R. Sikorski, Some examples of Borel sets, Colloq. Math. 5 (1958), 170–171. MR 104583, DOI 10.4064/cm-5-2-170-171
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 290 (1985), 1-39
- MSC: Primary 54H05; Secondary 04A15, 54E35, 54F65
- DOI: https://doi.org/10.1090/S0002-9947-1985-0787953-3
- MathSciNet review: 787953