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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Retractions and other continuous maps from $\beta X$ onto $\beta X_X$
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by W. W. Comfort PDF
Trans. Amer. Math. Soc. 114 (1965), 1-9 Request permission

Abstract:

Our two main theorems are stated below. The first is proved with the aid of the continuum hypothesis. Theorem 2.6. [CH] Suppose that there is a retraction from $\beta X$ onto $\beta X\backslash X$. Then X is locally compact and pseudocompact. Theorem 4.2. Let D be a discrete space whose cardinal number m exceeds 1. In order that there exist a continuous function from $\beta D$ onto $\beta D\backslash D$, it is necessary and sufficient that $\mathfrak {m} = {\mathfrak {m}^{\aleph _0}}$. The proof of Theorem 2.6 rests on a result of Walter Rudin concerning P-points (see 1(d) and 1(e) below); Theorem 4.2 depends on the following simple result, which appears to be new. Theorem 4.1. Let D be the discrete space with cardinal number $\mathfrak {m} (\geqq {\aleph _0})$. The smallest cardinal number which is the cardinal number of some dense subset of $\beta D\backslash D$ is ${\mathfrak {m}^{\aleph _0}}$.
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Additional Information
  • © Copyright 1965 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 114 (1965), 1-9
  • MSC: Primary 54.53
  • DOI: https://doi.org/10.1090/S0002-9947-1965-0185571-9
  • MathSciNet review: 0185571