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Transactions of the American Mathematical Society

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Retractions and other continuous maps from $ \beta X$ onto $ \beta X_X$

Author: W. W. Comfort
Journal: Trans. Amer. Math. Soc. 114 (1965), 1-9
MSC: Primary 54.53
MathSciNet review: 0185571
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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 [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1965 American Mathematical Society

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