Products and remote points: examples and counterexamples

Dow, A.; Peters, T. J.

doi:10.1090/S0002-9939-1988-0969061-X

Products and remote points: examples and counterexamples
HTML articles powered by AMS MathViewer

by A. Dow and T. J. Peters PDF

Proc. Amer. Math. Soc. 104 (1988), 1296-1304 Request permission

Abstract:

Examples of products with remote points and counterexamples of products without remote points are given. The paradoxical behavior of remote points with respect to products is exhibited. Also, an example is given of spaces $X$ and $Y$, where neither $X$ nor $Y$ has a $\sigma$-locally finite $\pi$-base, but $X \times Y$ does.

References

Soo Bong Chae and Jeffrey H. Smith, Remote points and $G$-spaces, Topology Appl. 11 (1980), no. 3, 243–246. MR 585269, DOI 10.1016/0166-8641(80)90023-1
W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267
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

Remote points

183

Eric K. van Douwen and Jan van Mill, Spaces without remote points, Pacific J. Math. 105 (1983), no. 1, 69–75. MR 688408
Alan Dow, Remote points in large products, Topology Appl. 16 (1983), no. 1, 11–17. MR 702616, DOI 10.1016/0166-8641(83)90003-2
Alan Dow, Some separable spaces and remote points, Canadian J. Math. 34 (1982), no. 6, 1378–1389. MR 678677, DOI 10.4153/CJM-1982-096-6
Alan Dow, Weak $P$-points in compact CCC $F$-spaces, Trans. Amer. Math. Soc. 269 (1982), no. 2, 557–565. MR 637709, DOI 10.1090/S0002-9947-1982-0637709-4
Alan Dow, Remote points in spaces with $\pi$-weight $\omega _1$, Fund. Math. 124 (1984), no. 3, 197–205. MR 774511, DOI 10.4064/fm-124-3-197-205
Alan Dow, Products without remote points, Topology Appl. 15 (1983), no. 3, 239–246. MR 694543, DOI 10.1016/0166-8641(83)90054-8

A separable space with no remote points

A. Dow and T. Peters, Game strategies yield remote points, Topology Appl. 27 (1987), no. 3, 245–256. MR 918534, DOI 10.1016/0166-8641(87)90089-7
N. J. Fine and L. Gillman, Remote points in $\beta R$, Proc. Amer. Math. Soc. 13 (1962), 29–36. MR 143172, DOI 10.1090/S0002-9939-1962-0143172-5
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
Stephen H. Hechler, On the existence of certain cofinal subsets of $^{\omega }\omega$, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1974, pp. 155–173. MR 0360266
M. Henriksen and T. J. Peters, Locally finite families, completely separated sets and remote points, Proc. Amer. Math. Soc. 103 (1988), no. 3, 989–995. MR 947695, DOI 10.1090/S0002-9939-1988-0947695-6
Gert H. Müller and Dana S. Scott (eds.), Higher set theory, Lecture Notes in Mathematics, vol. 669, Springer, Berlin, 1978. MR 520182
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
Kenneth Kunen and Franklin D. Tall, Between Martin’s axiom and Souslin’s hypothesis, Fund. Math. 102 (1979), no. 3, 173–181. MR 532951, DOI 10.4064/fm-102-3-173-181

More on remote points

Remote points, products and

-spaces

Thomas J. Peters, Dense homeomorphic subspaces of $X^\ast$ and of $(EX)^\ast$, Proceedings of the 1983 topology conference (Houston, Tex., 1983), 1983, pp. 285–301. MR 765084
Thomas J. Peters, For any $X$, the product $X\times Y$ has remote points for some $Y$, Proc. Amer. Math. Soc. 95 (1985), no. 4, 641–648. MR 810178, DOI 10.1090/S0002-9939-1985-0810178-5
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
Stevo Todorčević, Stationary sets, trees and continuums, Publ. Inst. Math. (Beograd) (N.S.) 29(43) (1981), 249–262. MR 657114
Russell C. Walker, The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698
Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581