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A separable space with no remote points


Author: Alan Dow
Journal: Trans. Amer. Math. Soc. 312 (1989), 335-353
MSC: Primary 54D35; Secondary 03E35, 03E55, 54A35, 54D40, 54D60
DOI: https://doi.org/10.1090/S0002-9947-1989-0983872-1
MathSciNet review: 983872
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Abstract: In the model obtained by adding $ {\omega _2}$ side-by-side Sacks reals to a model of $ {\mathbf{CH}}$, there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size $ {\omega _2}$ such that every uncountable subset has density $ {\omega _1}$. Furthermore assuming the existence of a measurable cardinal $ \kappa $ with $ {2^\kappa } = {\kappa ^ + }$, a space $ X$ is produced with no isolated points but with remote points in $ \upsilon X - X$. It is also shown that a pseudocompact space does not have remote points.


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  • [B1] J. E. Baumgartner, Results and independence proofs in combinatorial set theory, Ph.D. thesis, Univ. of Calif., Berkeley, 1970.
  • [B2] -, Sacks forcing and the total failure of Martin's Axiom, preprint.
  • [B3] -, Iterated forcing, Surveys in Set Theory, edited by A. R. D. Mathias, Cambridge Univ. Press, 1983. MR 823774 (86m:03005)
  • [BL] J. E. Baumgartner and R. Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), 271-288. MR 556894 (81a:03050)
  • [Be] M. G. Bell, Compact ccc non-separable spaces of small weight, Topology Proc. 5 (1980), 11-25. MR 624458 (83f:54027)
  • [Br] R. Blair, A note on remote points, preprint. MR 842308 (87g:54059)
  • [CS] S. B. Chae and J. H. Smith, Remote points and $ G$-spaces, Topology Appl. 11 (1980), 243-246. MR 585269 (81m:54037)
  • [ChKe] C. C. Chang and H. J. Keisler, Model theory, North-Holland, New York, 1977. MR 0532927 (58:27177)
  • [vD1] E. K. van Douwen, Existence and applications of remote points, Bull. Amer. Math. Soc. 841 (1978), 161-163. MR 0461463 (57:1448)
  • [vD2] -, Remote points, Dissertationes Math. 188 (1981). MR 627526 (83i:54024)
  • [vD3] -, Covering and separation properties of box products, Surveys in General Topology, edited by G. M. Reed, Academic Press, 1980, pp. 55-130. MR 564100 (81i:54006)
  • [vDvM] E. K. van Douwen and J. van Mill, Spaces without remote points, Pacific J. Math. 105 (1983), 69-75. MR 688408 (84d:54046)
  • [D1] A. Dow, Weak $ P$-points in compact ccc $ F$-spaces, Trans. Amer. Math. Soc. 269 (1982), 557-565. MR 637709 (83b:54028)
  • [D2] -, Some separable spaces and remote points, Canad. J. Math. 34 (1982), 1378-1389. MR 678677 (84b:54010)
  • [D3] -, Products without remote points, Topology Appl. 15 (1983), 239-246. MR 694543 (84f:54031)
  • [D4] -, Remote points in spaces with $ \pi $-weight $ {\omega _1}$, Fund. Math. 124 (1984), 197-205. MR 774511 (86d:54033)
  • [D5] -, Some linked subsets of posets, Israel J. Math. 59 (1987), 353-376. MR 920500 (88m:03073)
  • [DP] A. Dow and T. J. Peters, Game strategies yield remote points, Topology Appl. 27 (1987), 245-256. MR 918534 (89e:54045)
  • [DvM] A. Dow and J. van Mill, On nowhere dense ccc $ P$-sets, Proc. Amer. Math. Soc. 80 (1980), 697-700. MR 587958 (82a:54032)
  • [DTW] A. Dow, F. D. Tall and W. A. R. Weiss, New proofs of the normal Moore space conjecture, preprint.
  • [F1] W. G. Fleissner, The normal Moore space conjecture and large cardinals, Handbook of Set-Theoretic Topology, edited by K. Kunen and J. E. Vaughan, North-Holland, Amsterdam, 1984, pp. 733-760. MR 776635 (86m:54023)
  • [F2] -, Homomorphism axioms and lynxes, Axiomatic Set Theory, Edited by J. E. Baumgartner, D. A. Martin and S. Shelah, Contemp. Math., vol. 31, Amer. Math. Soc., Providence, R.I., 1984, pp. 79-97. MR 763887 (85g:03004)
  • [FG] N. J. Fine and L. Gillman, Remote points in $ \beta R$, Proc. Amer. Math. Soc. 13 (1962), 29-36. MR 0143172 (26:732)
  • [G1] C. L. Gates, A study of remote points of metric spaces, Ph.D. thesis, Univ. of Kansas, 1973.
  • [G2] -, Some structural properties of the set of remote points of a metric space, Canad. J. Math. 32 (1980), 195-209. MR 559795 (82a:54044)
  • [G3] -, A characterization of co-absoluteness for a class of metric spaces, Proc. Amer. Math. Soc. 80 (1980), 499-504. MR 581014 (81h:54028)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, New York, 1976. MR 0407579 (53:11352)
  • [KaMa] A. Kanamori and M. Magidor, The evolution of large cardinals in set theory, Lecture Notes in Math., vol. 699, 1978, pp. 99-275. MR 520190 (80b:03083)
  • [K1] K. Kunen, Set theory, North-Holland, Amsterdam, 1980. MR 597342 (82f:03001)
  • [K2] -, Weak $ P$-points in $ {N^\ast}$, Colloq. Math. Soc. János Bolyai, no. 23, Budapest, 1978, pp. 741-749.
  • [KvMM] K. Kunen, J. van Mill and C. F. Mills, On nowhere dense closed $ P$-sets, Proc. Amer. Math. Soc. 78 (1980), 119-123. MR 548097 (80h:54029)
  • [vM1] J. van Mill, Weak $ P$-points in compact $ F$-spaces, Topology Proc. 4 (1979), 609-628. MR 598298 (82d:54042)
  • [vM2] -, Weak $ P$-points in Čech-Stone compactifications, Trans. Amer. Math. Soc. 273 (1982), 657-678. MR 667166 (83k:54026)
  • [P] T. J. Peters, Remote points, products and $ G$-spaces, Ph.D. thesis, Wesleyan Univ., 1982.
  • [PoW] J. Porter and R. G. Woods, Nowhere dense subsets of metric spaces with applications to Stone-Čech compactifications, Canad. J. Math. 24 (1972), 622-630. MR 0324665 (48:3015)
  • [R] W. Rudin, Homogeneity problems in the theory of Čech compactifications, Duke J. Math. 29 (1956), 409-419. MR 0080902 (18:324d)
  • [Ru] M. E. Rudin, Lectures on set-theoretic topology, CBMS Regional Conf. Ser. in Math., no. 23, Amer. Math. Soc., Providence, R.I., 1975. MR 0367886 (51:4128)
  • [S] S. Shelah, Proper forcing, Lecture Notes in Math., vol. 940, Springer-Verlag, 1982. MR 675955 (84h:03002)
  • [T] F. D. Tall, Normality versus collectionwise normality, Handbook of Set-Theoretic Topology, edited by K. Kunen and J. E. Vaughan, North-Holland, Amsterdam, 1984. MR 776634 (86m:54022)
  • [Te] T. Terada, On remote points in $ \upsilon X - X$, Proc. Amer. Math. Soc. 77 (1979), 264-266. MR 542095 (81b:54026)
  • [VWa] J. Vermeer and E. Wattel, Remote points, far points and homogeneity of $ {X^\ast}$, Topology Structure II, Math. Centre Tracts, 116, Math. Centrum, Amsterdam, 1979, pp. 285-290. MR 565848 (82a:54043)
  • [Wi] S. Williams, Box products, Handbook of Set-Theoretic Topology, edited by K. Kunen and J. E. Vaughan, North-Holland, Amsterdam, 1984. MR 776619 (85k:54001)
  • [W1] R. G. Woods, Some $ {\aleph _0}$-bounded subsets of Stone-Čech compactifications, Israel J. Math. 9 (1971), 250-256. MR 0278266 (43:3997)
  • [W2] -, A Boolean algebra of regular closed subsets of $ \beta X - X$, Trans. Amer. Math. Soc. 154 (1971), 23-36. MR 0270341 (42:5230)
  • [W3] -, Homeomorphic sets of remote points, Canad. J. Math. 23 (1971), 495-502. MR 0281161 (43:6880)
  • [W4] -, Co-absolutes of remainders of Stone-Čech compactifications, Pacific J. Math. 37 (1971), 545-560. MR 0307179 (46:6300a)
  • [W5] -, A survey of absolutes of topological spaces, Topology Structure II, Math. Centre Tracts, 116, Math. Centrum, Amsterdam, 1979, pp. 323-362. MR 565852 (81d:54019)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0983872-1
Keywords: Remote points, measurable cardinals, side-by-side Sacks forcing
Article copyright: © Copyright 1989 American Mathematical Society

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