On 𝐶*-embedding in 𝛽𝑁 and the continuum hypothesis

Dow, Alan S.; Woods, R. Grant

doi:10.1090/S0002-9939-1985-0774021-5

On $C^ \ast$-embedding in $\beta \textbf {N}$ and the continuum hypothesis
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by Alan S. Dow and R. Grant Woods

Proc. Amer. Math. Soc. 93 (1985), 549-554

DOI: https://doi.org/10.1090/S0002-9939-1985-0774021-5

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Abstract:

Let $\beta {\mathbf {N}}$ denote the Stone-Čech compactification of the natural numbers ${\mathbf {N}}$ with the discrete topology. It is shown that the continuum hypothesis holds iff for each pair $X$ and $Y$ of homeomorphic subspaces of $\beta {\mathbf {N}}$, $X$ is ${C^*}$-embedded in $\beta {\mathbf {N}}$ iff $Y$ is. Related questions concerning ${C^*}$-embedded subsets of $\beta {\mathbf {N}}$ are investigated assuming the hypothesis ${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$.

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