Александровские чтения-2016 | 22-26 мая 2016
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 Участник: Фролкина Ольга Дмитриевна
 Имя: Фролкина Ольга ДмитриевнаНазвание: Cantor set in $\mathbb R^3$ ambiently universal for a special family of Antoine Necklaces Город, страна: Москва Организация: Абстракт: Let $F$ be a family of sets in $\mathbb R^3$. A set $U\subset \mathbb R^3$ is called {\it ambiently universal set for the family $F$} if for each set $M\in F$ there exists a homeomorphism $h_M$ of $\mathbb R^3$ onto itself such that $h_M (M) \subset U$. Bothe showed that in $\mathbb R^3$ there does not exist a closed zero-dimensional set ambiently universal for all Cantor sets [1]. Proof of Theorem 5.1 in [3] implies that in $\mathbb R^3$ there does not exist a closed zero-dimensional set ambiently universal for all Antoine necklaces. (By Antoine necklace we mean Cantor sets in $\mathbb R^3$ of special type; they generalize classical Antoine's construction and were defined in [2].) In the talk we will define a special class of Antoine necklaces and construct a Cantor set ambiently universal for this family. {\bf References} [1] H.G.Bothe, Zur Lage null- und eindimensionaler Punktmengen // Fund. Math. LVIII (1966), 1--30. [2] R.B.Sher, Concerning wild Cantor sets in $E^3$ // Proc. Amer. Math. Soc. 19 (1968), 1195-1200. [3] D.G. Wright, Ambiently universal sets in $E^n$ // Trans. Amer. Math. Soc. 277, 2 (1983), 655--664.

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