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Имя: Фролкина Ольга Дмитриевна Название: 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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