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, Volume 22, Issue 1, pp 77–85 | Cite as

Uncountable powers of\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R}\) can be almost Lindelöf

  • D. H. Fremlin


Moran ([6]) and, independently, Kemperman & Maharam ([4]) have shown that\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R} ^\mathbb{C}\) is not almost Lindelöf. Hechler ([3]) showed that\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R} ^{\aleph _1 }\) may fail to be almost Lindelöf whether or not the continuum hypothesis is true. In this paper I shall prove that, if Martin's Axiom is assumed, then\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R} ^K\) is almost Lindelöf for every k<ℂ; so that, in particular, it is possible for\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R} ^{\aleph _1 }\) to be almost Lindelöf. The novel part of the argument is contained in Theorem 7, which does not depend on Martin's Axiom.


Number Theory Algebraic Geometry Topological Group Continuum Hypothesis Uncountable Power 
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  1. 1.
    FREMLIN, D.H.: Topological Riesz Spaces and Measure Theory. Cambridge: University Press 1974.Google Scholar
  2. 2.
    FREMLIN, D.H., GARLING, D.J.H., HAYDON, R.G.: Bounded measures on topological spaces. Proc. London Math. Soc. (3)25, 115–136 (1972).Google Scholar
  3. 3.
    HECHLER, S.H.: On\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{N} ^{\aleph _1 }\) and the almost-Lindelöf property. Proc. Amer. Math. Soc.52, 353–355 (1975).Google Scholar
  4. 4.
    KEMPERMAN, J.H.B., MAHARAM, D.:\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R} ^\mathbb{C}\) is not almost Lindelöf. Proc. Amer. Math. Soc.24, 772–773 (1970).Google Scholar
  5. 5.
    MARTIN, D.A., SOLOVAY, R.M.: Internal Cohen extensions. Ann. Math. Logic2, 143–178 (1970).Google Scholar
  6. 6.
    MORAN, W.: The additivity of measures on completely regular spaces. J. London Math. Soc.43, 633–639 (1968).Google Scholar
  7. 7.
    ROSS, K.A., STONE, A.H.: Products of separable spaces. Amer. Math. Monthly71, 393–403 (1964).Google Scholar
  8. 8.
    RUDIN, M.E.: Martin's Axiom. Chapter B6 of Handbook of Mathematical Logic, ed. J.Barwise. North-Holland 1977.Google Scholar
  9. 9.
    SCHWARTZ, L.: Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford: University Press 1973.Google Scholar
  10. 10.
    SOLOVAY, R., TENNENBAUM, S.: Iterated Cohen extensions and Souslin's problem. Ann. of Math. (2)94, 201–245 (1971).Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • D. H. Fremlin
    • 1
  1. 1.Department of MathematicsUniversity of EssexColchesterEngland

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