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Menger curve

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An example of a curve containing the topological image of any curve (and, in addition, of every one-dimensional separable metrizable space). For this reason it is referred to as a universal curve. It was constructed by K. Menger [1] (for Menger's construction see Line (curve)). The Menger curve is topologically characterized [3] as a one-dimensional locally connected metrizable continuum $K$ without locally separating points (i.e. for every connected neighbourhood $O$ of any point $x \in K$ the set $O\setminus\{x\}$ is connected) and also without non-empty open subsets imbeddable in the plane.

References

[1] K. Menger, "Kurventheorie" , Teubner (1932)
[2] A.S. Parkhomenko, "What kind of curve is that?" , Moscow (1954) (In Russian)
[3] R. Anderson, "One-dimensional continuous curves and a homogeneity theorem" Ann. of Math. , 68 (1958) pp. 1–16
How to Cite This Entry:
Menger curve. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Menger_curve&oldid=43032
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article