# Menger curve

From Encyclopedia of Mathematics

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