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Keywords:
$\beta \mathbb{N}$; retracts; two to one map; Stone-Čech compactification
$\beta \mathbb{N}$; retracts; two to one map; Stone-Čech compactification
Summary:
Eric van Douwen produced in 1993 a maximal crowded extremally disconnected regular space and showed that its Stone-Čech compactification is an at most two-to-one image of $\beta {\mathbb{N}}$. We prove that there are non-homeomorphic such images. We also develop some related properties of spaces which are absolute retracts of $\beta {\mathbb{N}}$ expanding on earlier work of Balcar and Błaszczyk (1990) and Simon (1987).
References:
[1] A. Bella, A. Błaszczyk, A. Szymański: On absolute retracts of $\omega ^\ast $. Fund. Math. 145 (1994), 1–13. DOI 10.4064/fm-145-1-1-13 | MR 1295157
[2] B. Balcar, A. Błaszczyk: On minimal dynamical systems on Boolean algebras. Comment. Math. Univ. Carolin. 31 (1990), 7–11. MR 1056164
[3] Eric K. van Douwen: Applications of maximal topologies. Topology Appl. 51 (1993), 125–139. DOI 10.1016/0166-8641(93)90145-4 | MR 1229708
[4] A. Dow, A. V. Gubbi, A. Szymański: Rigid Stone spaces within ${\mathrm ZFC}$. Proc. Amer. Math. Soc. 102 (1988), 745–748. MR 0929014
[5] Alan Dow: $\beta {\mathbb{N}}$. The work of Mary Ellen Rudin (Madison, WI, 1991), Ann. New York Acad. Sci., vol. 705, New York Acad. Sci., New York, 1993, pp. 47–66. MR 1277880
[6] A. Dow, J. van Mill: On $n$-to-one continuous images of $\beta {\mathbb{N}}$. Preprint 2005. MR 2361681
[7] Lutz Heindorf, Leonid B. Shapiro: Nearly projective Boolean algebras. Lecture Notes Math., vol. 1596, Springer, Berlin, 1994. With an appendix by Sakaé Fuchino. DOI 10.1007/BFb0094103 | MR 1329090
[8] Neil Hindman, Dona Strauss: Recent progress in the topological theory of semigroups and the algebra of $\beta S$. Recent progress in general topology, II, North-Holland, Amsterdam, 2002, pp. 227–251. MR 1970000
[9] S. Koppelberg: Characterizations of Cohen algebras. Papers on General Topology and Applications (Madison, WI, 1991), Ann. New York Acad. Sci., vol. 704, New York Acad. Sci., New York, 1993, pp. 222–237. MR 1277859 | Zbl 0830.06006
[10] R. Levy: The weight of certain images of $\omega ^{*}$. Topology Appl. 153 (2006), 2272–2277. DOI 10.1016/j.topol.2004.06.015 | MR 2238730 | Zbl 1099.54016
[11] Jan van Mill: An Introduction to $\beta \omega $. Handbook of Set-Theoretic Topology, K. Kunen, J. E. Vaughan (eds.), Elsevier Science Publishers BV, North-Holland, Amsterdam, 1984, pp. 503–567. MR 0776630
[12] L. B. Shapiro: On spaces that are coabsolute with dyadic compacta. Dokl. Akad. Nauk SSSR 293 (1987), 1077–1081. MR 0890202
[13] P. Simon: A closed separable subspace of $\beta {\mathbb{N}}$ which is not a retract. Trans. Amer. Math. Soc. 299 (1987), 641–655. MR 0869226
[14] A. Szymański: Some applications of tiny sequences. Proceedings of the 11th Winter School on Abstract Analysis (Železná Ruda, 1983), Suppl. 3, 1984, pp. 321–328. MR 0744396
[15] M. Talagrand: Non existence de relèvement pour certaines mesures finiement additives et retractés de $\beta {\mathbb{N}}$. Math. Ann. 256 (1981), 63–66. DOI 10.1007/BF01450944 | MR 0620123
[16] Jerry E. Vaughan: Two spaces homeomorphic to ${\mathrm Seq}(p)$. Comment. Math. Univ. Carolin. 42 (2001), 209–218. MR 1825385
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