Products of $\omega ^*$ images
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- by M. Bell, L. Shapiro and P. Simon PDF
- Proc. Amer. Math. Soc. 124 (1996), 1593-1599 Request permission
Abstract:
Let $\omega ^*$ be the C̆ech-Stone remainder $\beta \omega \setminus \omega$. We show that there exists a large class $\mathcal {O}$ of images of $\omega ^*$ such that whenever $\mathcal {S}$ is a subset of $\mathcal {O}$ of cardinality at most the continuum, then $\omega ^* \times \prod \mathcal {S}$ is again an image of $\omega ^*$. The class $\mathcal {O}$ contains all separable compact spaces, all compact spaces of weight at most $\omega _1$ and all perfectly normal compact spaces.References
- Murray G. Bell, A first countable compact space that is not an $N^*$ image, Topology Appl. 35 (1990), no. 2-3, 153–156. MR 1058795, DOI 10.1016/0166-8641(90)90100-G
- Aleksander Błaszczyk and Andrzej Szymański, Concerning Parovičenko’s theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 7-8, 311–314 (1981) (English, with Russian summary). MR 628044
- Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
- Eric K. van Douwen and Teodor C. Przymusiński, Separable extensions of first countable spaces, Fund. Math. 105 (1979/80), no. 2, 147–158. MR 561588, DOI 10.4064/fm-105-2-147-158
- Winfried Just, The space $(\omega ^*)^{n+1}$ is not always a continuous image of $(\omega ^*)^n$, Fund. Math. 132 (1989), no. 1, 59–72. MR 1004296, DOI 10.4064/fm-132-1-59-72
- M. A. Maurice, Compact ordered spaces, Mathematical Centre Tracts, vol. 6, Mathematisch Centrum, Amsterdam, 1964. MR 0220252
- Jan van Mill, An introduction to $\beta \omega$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR 776630
- I.Parovic̆enko, A Universal Bicompact Of Weight $\aleph$, Soviet Math. Doklady 4, (1963), 592-595.
- Teodor C. Przymusiński, Perfectly normal compact spaces are continuous images of $\beta \textbf {N}\sbs \textbf {N}$, Proc. Amer. Math. Soc. 86 (1982), no. 3, 541–544. MR 671232, DOI 10.1090/S0002-9939-1982-0671232-1
- B.S̆apirovski, Canonical Sets And Character. Density And Weight Of Bicompacta, Dokl. Acad. Nauk SSSR 218, (1974), 58-61.
Additional Information
- M. Bell
- Affiliation: Department of Mathematics, University of Manitoba, Fort Garry Campus, Winnipeg, Canada R3T 2N2
- Email: mbell@cc.umanitoba.ca
- L. Shapiro
- Affiliation: Department of Mathematics, Academy of Labor and Social Relations, Lobachevskogo 90, Moscow, Russia 117454
- Email: lshapiro@glas.apc.org
- P. Simon
- Affiliation: Matematický Ústav, University Karlovy, Sokolovská 83, 18600 Praha 8, Czech Republic
- Email: psimon@ms.mff.cuni.cz
- Received by editor(s): October 20, 1994
- Additional Notes: The first author gratefully acknowledges support from NSERC of Canada. The second author collaborated while visiting the University of Manitoba, Canada and also thanks the International Science Foundation for support. The third author gratefully acknowledges support by Charles University grant GAUK 350. We would like to thank A. Dow for helpful communications; in particular, for showing us his proof that $\omega _1 + 1$ is an orthogonal $\omega ^*$ image.
- Communicated by: Franklin D. Tall
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1593-1599
- MSC (1991): Primary 54D30, 06E05; Secondary 54B10, 54D40
- DOI: https://doi.org/10.1090/S0002-9939-96-03385-0
- MathSciNet review: 1328339