On complete metric spaces containing the Sierpinski curve

Prajs, Janusz

doi:10.1090/S0002-9939-98-04509-2

On complete metric spaces containing the Sierpinski curve
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by Janusz R. Prajs PDF

Proc. Amer. Math. Soc. 126 (1998), 3743-3747 Request permission

Abstract:

It is proved that a complete metric space topologically contains the Sierpiński 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 $A\setminus \{a\}$.

References

R. D. Anderson, A characterization of the universal curve and a proof of its homogeneity, Ann. of Math. (2) 67 (1958), 313–324. MR 96180, DOI 10.2307/1970007
R. D. Anderson, One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. (2) 68 (1958), 1–16. MR 96181, DOI 10.2307/1970040
P. Krupski and H. Patkowska, Menger curves in Peano continua, Colloq. Math. 70 (1996), 79–86.
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
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
G. T. Whyburn, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958), 320–324. MR 99638, DOI 10.4064/fm-45-1-320-324