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 | References | Similar Articles | Additional Information

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 containing an arc such that for each and for each open arc with , there exists an arbitrary small arc in joining the two components of .

**1.**R. D. Anderson,*A characterization of the universal curve and a proof of its homogeneity*, Ann. of Math. (2)**67**(1958), 313–324. MR**0096180****2.**R. D. Anderson,*One-dimensional continuous curves and a homogeneity theorem*, Ann. of Math. (2)**68**(1958), 1–16. MR**0096181****3.**P. Krupski and H. Patkowska,*Menger curves in Peano continua,*Colloq. Math. 70 (1996), 79-86. CMP**96:08****4.**J. C. Mayer, Lex G. Oversteegen, and E. D. Tymchatyn,*The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets*, Dissertationes Math. (Rozprawy Mat.)**252**(1986), 45. MR**874232****5.**Gordon Thomas Whyburn,*Analytic Topology*, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. MR**0007095****6.**G. T. Whyburn,*Topological characterization of the Sierpiński curve*, Fund. Math.**45**(1958), 320–324. MR**0099638**

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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:
https://doi.org/10.1090/S0002-9939-98-04509-2

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