Retractions and other continuous maps from $\beta X$ onto $\beta X_X$
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- by W. W. Comfort
- Trans. Amer. Math. Soc. 114 (1965), 1-9
- DOI: https://doi.org/10.1090/S0002-9947-1965-0185571-9
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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 ${\mathfrak {m}^{\aleph _0}}$.References
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Bibliographic 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