Ambiently universal sets in 𝐸ⁿ

Wright, David G.

doi:10.1090/S0002-9947-1983-0694381-6

Ambiently universal sets in $E^{n}$
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by David G. Wright

Trans. Amer. Math. Soc. 277 (1983), 655-664

DOI: https://doi.org/10.1090/S0002-9947-1983-0694381-6

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Abstract:

For each closed set $X$ in ${E^n}$ of dimension at most $n - 3$, we show that $X$ fails to be ambiently universal with respect to Cantor sets in ${E^n}$; i.e., we find a Cantor set $Y$ in ${E^n}$ so that for any self-homeomorphism $h$ of ${E^n}$, $h(Y)$ is not contained in $X$. This result answers a question posed by H. G. Bothe and completes the understanding of ambiently universal sets in ${E^n}$.

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There is no universal Cantor set in