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

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

MathSciNet review:
1458258

Full-text PDF

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 the proof of its homogeneity,*Ann. of Math. 67 (1958), 313-324. MR**20:2675****2.**R. D. Anderson,*One-dimensional continuous curves and a homogeneity theorem,*Ann. of Math. 68 (1958), 1-16. MR**20:2676****3.**P. Krupski and H. Patkowska,*Menger curves in Peano continua,*Colloq. Math. 70 (1996), 79-86. CMP**96:08****4.**J C. Mayer, L. G. Oversteegen and E. D. Tymchatyn,*The Menger curve characterization and extension of homeomorphisms of non-locally-separating closed subsets,*Dissert. Math. 252 (1986), 1-45. MR**87m:54106****5.**G. T. Whyburn,*Analytic topology,*Amer. Math. Soc. Colloq. Publ., vol.28, Amer. Math. Soc., Providence, R. I., 1942. MR**4:86b****6.**G. T. Whyburn,*Topological characterization of the Sierpi\'{n}ski curve,*Fund. Math. 45 (1958), 320-324. MR**20:6077**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
54F15,
54F65,
54F50,
54C25

Retrieve articles in all journals with MSC (1991): 54F15, 54F65, 54F50, 54C25

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