For any $X$, the product $X\times Y$ has remote points for some $Y$
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- by Thomas J. Peters PDF
- Proc. Amer. Math. Soc. 95 (1985), 641-648 Request permission
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.References
- Soo Bong Chae and Jeffrey H. Smith, Remote points and $G$-spaces, Topology Appl. 11 (1980), no.ย 3, 243โ246. MR 585269, DOI 10.1016/0166-8641(80)90023-1
- W. W. Comfort, A survey of cardinal invariants, General Topology and Appl. 1 (1971), no.ย 2, 163โ199. MR 290326, DOI 10.1016/0016-660X(71)90122-X
- W. W. Comfort, Products of spaces with properties of pseudocompactness type, Topology Proc. 4 (1979), no.ย 1, 51โ65 (1980). MR 583688
- W. W. Comfort, Ultrafilters: an interim report, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp.ย 33โ54. MR 564099
- W. Wistar Comfort and Stylianos A. Negrepontis, Chain conditions in topology, Cambridge Tracts in Mathematics, vol. 79, Cambridge University Press, Cambridge-New York, 1982. MR 665100, DOI 10.1017/CBO9780511897337 W. W. Comfort, Personal communication, May 6, 1982. E. K. van Douwen, Remote points, Dissertationes Math. 183 (1982).
- Eric K. van Douwen and Jan van Mill, Spaces without remote points, Pacific J. Math. 105 (1983), no.ย 1, 69โ75. MR 688408, DOI 10.2140/pjm.1983.105.69
- Alan Dow, Remote points in large products, Topology Appl. 16 (1983), no.ย 1, 11โ17. MR 702616, DOI 10.1016/0166-8641(83)90003-2
- Alan Dow, Products without remote points, Topology Appl. 15 (1983), no.ย 3, 239โ246. MR 694543, DOI 10.1016/0166-8641(83)90054-8
- Zdenฤk Frolรญk, Non-homogeneity of $\beta P-P$, Comment. Math. Univ. Carolinae 8 (1967), 705โ709. MR 266160
- N. J. Fine and L. Gillman, Remote points in $\beta R$, Proc. Amer. Math. Soc. 13 (1962), 29โ36. MR 143172, DOI 10.1090/S0002-9939-1962-0143172-5
- Catherine L. Gates, Some structural properties of the set of remote points of a metric space, Canadian J. Math. 32 (1980), no.ย 1, 195โ209. MR 559795, DOI 10.4153/CJM-1980-015-7
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition. MR 0407579
- M. Henriksen and T. J. Peters, Locally finite families, completely separated sets and remote points, Proc. Amer. Math. Soc. 103 (1988), no.ย 3, 989โ995. MR 947695, DOI 10.1090/S0002-9939-1988-0947695-6
- I. Juhรกsz, Cardinal functions in topology, Mathematical Centre Tracts, No. 34, Mathematisch Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S. Kroonenberg. MR 0340021 J. van Mill, More on remote points, Rapport 91, Wiskundig Seminarium, Free University of Amsterdam, 1979. T. J. Peters, Remote points, products and $G$-spaces, Doctoral dissertation, Wesleyan University, Middletown, Connecticut, 1982.
- Thomas J. Peters, $G$-spaces: products, absolutes and remote points, Proceedings of the 1982 Topology Conference (Annapolis, Md., 1982), 1982, pp.ย 119โ146. MR 696626
- Thomas J. Peters, Dense homeomorphic subspaces of $X^\ast$ and of $(EX)^\ast$, Proceedings of the 1983 topology conference (Houston, Tex., 1983), 1983, pp.ย 285โ301. MR 765084
- Donald Plank, On a class of subalgebras of $C(X)$ with applications to $\beta XX$, Fund. Math. 64 (1969), 41โ54. MR 244953, DOI 10.4064/fm-64-1-41-54
- V. I. Ponomarev, Spaces co-absolute with metric spaces, Uspehi Mat. Nauk 21 (1966), no.ย 4 (130), 101โ132 (Russian). MR 0200902
- K. A. Ross and A. H. Stone, Products of separable spaces, Amer. Math. Monthly 71 (1964), 398โ403. MR 164314, DOI 10.2307/2313241
- M. G. Tkaฤenko, On $\pi$-bases of rank $1$ in infinite Cartesian products, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (1980), 52โ55, 108 (Russian, with English summary). MR 577945
- Stevo Todorฤeviฤ, Stationary sets, trees and continuums, Publ. Inst. Math. (Beograd) (N.S.) 29(43) (1981), 249โ262. MR 657114
- J. Vermeer and E. Wattel, Remote points, far points and homogeneity of $X^{\ast }$, Topological structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978) Math. Centre Tracts, vol. 116, Math. Centrum, Amsterdam, 1979, pp.ย 285โ290. MR 565848
- Russell C. Walker, The Stone-ฤech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698, DOI 10.1007/978-3-642-61935-9
- H. E. White Jr., First countable spaces that have special pseudo-bases, Canad. Math. Bull. 21 (1978), no.ย 1, 103โ112. MR 482615, DOI 10.4153/CMB-1978-016-5
- Scott W. Williams, Trees, Gleason spaces, and coabsolutes of $\beta \textbf {N}\sim \textbf {N}$, Trans. Amer. Math. Soc. 271 (1982), no.ย 1, 83โ100. MR 648079, DOI 10.1090/S0002-9947-1982-0648079-X
- R. Grant Woods, Homeomorphic sets of remote points, Canadian J. Math. 23 (1971), 495โ502. MR 281161, DOI 10.4153/CJM-1971-052-1
Additional Information
- © Copyright 1985 American Mathematical Society
- 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