The orderability and closed images of scattered spaces
HTML articles powered by AMS MathViewer
- by S. Purisch PDF
- Trans. Amer. Math. Soc. 320 (1990), 713-725 Request permission
Abstract:
A (totally) orderable scattered space and a space homeomorphic to a subspace of an ordinal space are characterized in terms of a neighborhood subbase for each of their points plus what corresponds to a neighborhood base for each of their non-$Q$-gaps. These generalize the characterizations in [ P$_{1}$] of an orderable compact scattered space and in [B] of a space homeomorphic to a compact ordinal space. Generalizing a result in [M] it is shown that a space is orderable and scattered iff it is the $2$ to $1$ image under a closed map of a subspace of an ordinal space. In response to a question of Telgarsky [T] a simple description is given of a closed map with discrete fibers from an orderable scattered space onto an orderable perfect space. Maps that preserve length conditions on a scattered space are touched upon.References
- John Warren Baker, Compact spaces homeomorphic to a ray of ordinals, Fund. Math. 76 (1972), no. 1, 19–27. MR 307197, DOI 10.4064/fm-76-1-19-27
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- R. Engelking and D. Lutzer, Paracompactness in ordered spaces, Fund. Math. 94 (1977), no. 1, 49–58. MR 428278, DOI 10.4064/fm-94-1-25-33
- V. V. Fedorčuk, Ordered spaces, Dokl. Akad. Nauk SSSR 169 (1966), 777–780 (Russian). MR 0208565
- Leonard Gillman and Melvin Henriksen, Concerning rings of continuous functions, Trans. Amer. Math. Soc. 77 (1954), 340–362. MR 63646, DOI 10.1090/S0002-9947-1954-0063646-5
- R. W. Heath, D. J. Lutzer, and P. L. Zenor, Monotonically normal spaces, Trans. Amer. Math. Soc. 178 (1973), 481–493. MR 372826, DOI 10.1090/S0002-9947-1973-0372826-2
- V. Kannan and M. Rajagopalan, Scattered spaces. II, Illinois J. Math. 21 (1977), no. 4, 735–751. MR 474180
- V. Kannan and M. Rajagopalan, Constructions and applications of rigid spaces. I, Adv. in Math. 29 (1978), no. 1, 89–130. MR 501093, DOI 10.1016/0001-8708(78)90006-3
- Gadi Moran, Order-two continuous Hausdorff images of compact ordinals, Israel J. Math. 42 (1982), no. 1-2, 75–86. MR 687936, DOI 10.1007/BF02765012
- S. Purisch, The orderability of compact scattered spaces, Topology and order structures, Part 2 (Amsterdam, 1981) Math. Centre Tracts, vol. 169, Math. Centrum, Amsterdam, 1983, pp. 53–61. MR 736695
- S. Purisch, Scattered compactifications and the orderability of scattered spaces. II, Proc. Amer. Math. Soc. 95 (1985), no. 4, 636–640. MR 810177, DOI 10.1090/S0002-9939-1985-0810177-3
- Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Wyoming, Laramie, Wyo., August 12–16, 1974. MR 0367886 V. Slucká, personal communication.
- R. Telgársky, Total paracompactness and paracompact dispersed spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 567–572 (English, with Russian summary). MR 235517
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 320 (1990), 713-725
- MSC: Primary 54F05; Secondary 54D99
- DOI: https://doi.org/10.1090/S0002-9947-1990-0989584-0
- MathSciNet review: 989584