Difference between revisions of "Menger curve"
From Encyclopedia of Mathematics
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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 [[#References|[1]]] (for Menger's construction see [[ | + | 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 [[#References|[1]]] (for Menger's construction see [[Line (curve)]]). The Menger curve is topologically characterized [[#References|[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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Menger, "Kurventheorie" , Teubner (1932)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.S. Parkhomenko, "What kind of curve is that?" , Moscow (1954) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R. Anderson, "One-dimensional continuous curves and a homogeneity theorem" ''Ann. of Math.'' , '''68''' (1958) pp. 1–16</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> K. Menger, "Kurventheorie" , Teubner (1932)</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> A.S. Parkhomenko, "What kind of curve is that?" , Moscow (1954) (In Russian)</TD></TR> | ||
| + | <TR><TD valign="top">[3]</TD> <TD valign="top"> R. Anderson, "One-dimensional continuous curves and a homogeneity theorem" ''Ann. of Math.'' , '''68''' (1958) pp. 1–16</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 18:18, 27 March 2018
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://encyclopediaofmath.org/index.php?title=Menger_curve&oldid=43032
Menger curve. Encyclopedia of Mathematics. URL: http://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