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

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 ${\mathfrak {m}^{\aleph _0}}$.