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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On complete metric spaces
containing the Sierpinski curve


Author: Janusz R. Prajs
Journal: Proc. Amer. Math. Soc. 126 (1998), 3743-3747
MSC (1991): Primary 54F15, 54F65, 54F50, 54C25
MathSciNet review: 1458258
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Abstract: It is proved that a complete metric space topologically contains the Sierpinski universal plane curve if and only if it has a subset with so-called bypass property, i.e. it has a subset $K$ containing an arc such that for each $a\in K$ and for each open arc $A\subset K$ with $a\in A$, there exists an arbitrary small arc in $K\setminus \{a\}$ joining the two components of $A\setminus \{a\}$.


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Additional Information

Janusz R. Prajs
Affiliation: Institute of Mathematics, Opole University, ul. Oleska 48, 45-052 Opole, Poland
Email: jrprajs@math.uni.opole.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04509-2
PII: S 0002-9939(98)04509-2
Keywords: Bypass property, embedding, homogeneity, local separating point, Sierpi\'nski curve
Received by editor(s): December 19, 1996
Received by editor(s) in revised form: April 21, 1997
Additional Notes: The author expresses grateful thanks to Prof. K. Omiljanowski for his help in the preparation of this paper.
Communicated by: Alan Dow
Article copyright: © Copyright 1998 American Mathematical Society