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For any $ X$, the product $ X\times Y$ has remote points for some $ Y$


Author: Thomas J. Peters
Journal: Proc. Amer. Math. Soc. 95 (1985), 641-648
MSC: Primary 54D40; Secondary 54A25, 54B10, 54B25, 54D35, 54G20
DOI: https://doi.org/10.1090/S0002-9939-1985-0810178-5
MathSciNet review: 810178
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Abstract: Any space with a $ \sigma $-locally finite $ \pi $-base will be called a $ \sigma - \pi $ space. The work of Chae and Smith can be extended to show that every nonpseudocompact $ \sigma - \pi $ space has remote points.$ ^{2}$ Sufficient conditions for a product to be a $ \sigma - \pi $ space are developed. It is shown that, for each space, if $ \alpha $ is a cardinal with the discrete topology, where $ \alpha $ is not less than $ \pi $-weight of $ X$, then $ X \times {\alpha ^\omega }$ has remote points. Cardinal function criteria are developed for the existence of $ \sigma - \pi $ spaces. An example is given of a pathological product which is a $ \sigma - \pi $ space even though none of its finite partial products is a $ \sigma - \pi $ space.


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  • [CS] S. B. Chae and J. H. Smith, Remote points and $ G$-spaces, Topology Appl. 11 (1980), 243-246. MR 585269 (81m:54037)
  • [C$ _{1}$] W. W. Comfort, A survey of cardinal invariants, Topology Appl. 1 (1971), 163-199. MR 0290326 (44:7510)
  • [C$ _{2}$] -, Products of spaces with properties of pseudo-compactness type, Topology Proc. 4 (1979), 51-65. MR 583688 (81j:54039)
  • [C$ _{3}$] -, Ultrafilters: An interim report, Surveys in General Topology (G. M. Reed, ed.), Academic Press, New York, 1980, pp. 33-54. MR 564099 (81a:54007)
  • [C$ _{4}$] W. W. Comfort and S. Negrepontis, Chain conditions in topology, Cambridge Univ. Press, Cambridge, 1982. MR 665100 (84k:04002)
  • [C$ _{5}$] W. W. Comfort, Personal communication, May 6, 1982.
  • [vD] E. K. van Douwen, Remote points, Dissertationes Math. 183 (1982).
  • [vDvM] E. K. van Douwen and J. van Mill, Spaces without remote points, Pacific J. Math. 105 (1983), 69-75. MR 688408 (84d:54046)
  • [D$ _{1}$] A. Dow, Remote points in large products, Topology Appl. 16 (1983), 11-17. MR 702616 (85d:54027)
  • [D$ _{2}$] -, Products without remote points, Topology Appl. 15 (1983), 239-246. MR 694543 (84f:54031)
  • [F] Z. Frolík, Nonhomogeneity of $ \beta P\backslash P$, Comment. Math. Univ. Carolinae 8 (1967), 705-709. MR 42, No. 1068. MR 0266160 (42:1068)
  • [FG] N. J. Fine and L. Gillman, Remote points in $ \beta {\mathbf{R}}$, Proc. Amer. Math. Soc. 13 (1962), 29-36. MR 26, No. 732. MR 0143172 (26:732)
  • [G] C. L. Gates, Some structural properties of the set of remote points of a metric space, Canad. J. Math. 32 (1980), 195-209. MR 559795 (82a:54044)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, New York, 1976. MR 0407579 (53:11352)
  • [HP] M. Henriksen and T. J. Peters, Locally finite families, completely separated sets and remote points, Preprint. MR 947695 (89e:54047)
  • [J] I. Juhász, Cardinal functions in topology, Mathematical Centre Tracts, no. 34, Mathematical Centrum, Amsterdam, 1971. MR 0340021 (49:4778)
  • [vM] J. van Mill, More on remote points, Rapport 91, Wiskundig Seminarium, Free University of Amsterdam, 1979.
  • [P$ _{1}$] T. J. Peters, Remote points, products and $ G$-spaces, Doctoral dissertation, Wesleyan University, Middletown, Connecticut, 1982.
  • [P$ _{2}$] -, $ G$-spaces: Products, absolutes, and remote points, Topology Proc. 7 (1982), 119-146. MR 696626 (84m:54027)
  • [P$ _{3}$] -, Dense homeomorphic subspaces of $ {X^ * }$ and of $ {\left( {EX} \right)^ * }$, Topology Proc. 8 (1983), 285-303. MR 765084 (86c:54024)
  • [P$ _{1}$] D. Plank, On a class of subalgebras of $ C\left( X \right)$ with applications to $ \beta X\backslash X$, Fund. Math. 64 (1969), 41-54. MR 0244953 (39:6266)
  • [Po] V. I. Ponomarev, On spaces co-absolute with metric spaces, Uspekhi Mat. Nauk 21(4) (1966), 101-132 [English transl., Russian Math. Surveys 21(4), (1966), 87-114]. MR 0200902 (34:788)
  • [RS] K. A. Ross and A. H. Stone, Products of separable spaces, Amer. Math. Monthly 71 (1964), 398-403. MR 0164314 (29:1611)
  • [T] M. G. Tkachenko, $ \pi $-bases of rank 1 in infinite products, Vestnik Moskov. Mat. 35 (1980), 52-55 [English transl., Moscow Univ. Math. Bull. 35 (1980), 55-58]. MR 577945 (81h:54006)
  • [To] S. Todorcevic, Stationary sets, trees and continuums, Publications de L'Institut Mathematique 41 (1981), 249-262. MR 657114 (84g:03078)
  • [VW] J. Vermeer and E. Wattel, Remote points, far points and homogeneity of $ {X^ * }$, Topological Structures II, Part 2, Proc. Sympos., Amsterdam, 1978, Math. Centre Tracts, no. 116, Amsterdam, 1979, pp. 285-290. MR 565848 (82a:54043)
  • [W] R. C. Walker, The Stone-Čech Compactification, Springer-Verlag, New York, 1974. MR 0380698 (52:1595)
  • [Wh] H. E. White, First countable spaces having special pseudobases, Canad. Math. Bull. 21 (1978), 103-112. MR 0482615 (58:2675)
  • [Wi] S. W. Williams, Trees, Gleason spaces and coabsolutes of $ \beta {\mathbf{N}} - {\mathbf{N}}$, Trans. Amer. Math. Soc. 271 (1982), 83-100. MR 648079 (83d:54060)
  • [Wo] R. G. Woods, Homeomorphic sets of remote points, Canad. J. Math. 23 (1971), 495-502. MR 0281161 (43:6880)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0810178-5
Keywords: $ \sigma - \pi $ space, products, $ G$-space, $ \sigma $-locally finite $ \pi $-base, remote point, Stone-Čech compactification, remainder, homogeneity, pseudo-$ \gamma $-compact
Article copyright: © Copyright 1985 American Mathematical Society

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