The influence of a small cardinal on the product of a Lindelöf space and the irrationals

Author:
L. Brian Lawrence

Journal:
Proc. Amer. Math. Soc. **110** (1990), 535-542

MSC:
Primary 54B10; Secondary 03E35, 54D20

DOI:
https://doi.org/10.1090/S0002-9939-1990-1021211-4

MathSciNet review:
1021211

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Abstract | References | Similar Articles | Additional Information

Abstract: It is unknown whether there is in ZFC a Lindelöf space whose product with the irrationals is nonnormal. We give some necessary conditions based on the minimum cardinality of unbounded family in .

**[1]**K. Alster,*The product of a Lindelöf space with the space of irrationals under Martin’s axiom*, Proc. Amer. Math. Soc.**110**(1990), no. 2, 543–547. MR**993736**, https://doi.org/10.1090/S0002-9939-1990-0993736-9**[2]**A. S. Besicovitch,*Concentrated and rarified sets of points*, Acta Math.**62**(1933), no. 1, 289–300. MR**1555386**, https://doi.org/10.1007/BF02393607**[3]**D. K. Burke and S. W. Davis,*Subsets of ^{𝜔}𝜔 and generalized metric spaces*, Pacific J. Math.**110**(1984), no. 2, 273–281. MR**726486****[4]**Ryszard Engelking,*General topology*, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR**0500780****[5]**Stephen H. Hechler,*On the existence of certain cofinal subsets of ^{𝜔}𝜔*, 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****[6]**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****[7]**Kenneth Kunen and Jerry E. Vaughan (eds.),*Handbook of set-theoretic topology*, North-Holland Publishing Co., Amsterdam, 1984. MR**776619****[8]**L. B. Lawrence,*Lindelöf spaces concentrated on Bernstein subsets of the real line*(to appear).**[9]**E. Michael,*The product of a normal space and a metric space need not be normal*, Bull. Amer. Math. Soc.**69**(1963), 375–376. MR**0152985**, https://doi.org/10.1090/S0002-9904-1963-10931-3**[10]**Ernest A. Michael,*Paracompactness and the Lindelöf property in finite and countable Cartesian products*, Compositio Math.**23**(1971), 199–214. MR**0287502****[11]**John C. Oxtoby,*Measure and category*, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR**584443****[12]**Mary Ellen Rudin,*Countable paracompactness and Souslin’s problem*, Canad. J. Math.**7**(1955), 543–547. MR**0073155**, https://doi.org/10.4153/CJM-1955-058-8**[13]**-,*A normal space**for which**is not normal*, Fund. Math.**73**(1971), 179-186.**[14]**Mary Ellen Rudin and Michael Starbird,*Products with a metric factor*, General Topology and Appl.**5**(1975), no. 3, 235–248. MR**0380709**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1021211-4

Keywords:
Product,
irrationals,
separable completely metrizable space,
Lindelöf space,
normal,
Michael line,
concentrated space,
Continuum Hypothesis,
Martin's axiom

Article copyright:
© Copyright 1990
American Mathematical Society