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A metrizable one-dimensional continuum. A Cantor curve originally referred to a plane nowhere-dense continuum, and it was the first (but not intrinsic) characterization of one-dimensional closed connected subsets of the plane considered by G. Cantor. A Cantor curve contains a nowhere-dense subcontinuum if and only if the closure of the set of its branching points is one-dimensional. If, on the other hand, a Cantor curve does not contain a nowhere-dense subcontinuum, then all its points have finite branch index. A Cantor curve without branching points is either a simple arc or a simple closed line. The set of end points of a Cantor curve, i.e. the set of points of index 1, is zero-dimensional, but can be everywhere dense. If all points of a Cantor curve have the same finite branch index, then the Cantor curve is a simple closed line. The universal Cantor curve (the Menger curve) can be constructed; this is a Cantor curve that contains a topological image of every Cantor curve.
 
A metrizable one-dimensional continuum. A Cantor curve originally referred to a plane nowhere-dense continuum, and it was the first (but not intrinsic) characterization of one-dimensional closed connected subsets of the plane considered by G. Cantor. A Cantor curve contains a nowhere-dense subcontinuum if and only if the closure of the set of its branching points is one-dimensional. If, on the other hand, a Cantor curve does not contain a nowhere-dense subcontinuum, then all its points have finite branch index. A Cantor curve without branching points is either a simple arc or a simple closed line. The set of end points of a Cantor curve, i.e. the set of points of index 1, is zero-dimensional, but can be everywhere dense. If all points of a Cantor curve have the same finite branch index, then the Cantor curve is a simple closed line. The universal Cantor curve (the Menger curve) can be constructed; this is a Cantor curve that contains a topological image of every Cantor curve.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Urysohn,  "Works on topology and other areas of mathematics" , '''2''' , Moscow-Leningrad  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Menger,  "Kurventheorie" , Teubner  (1932)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Urysohn,  "Works on topology and other areas of mathematics" , '''2''' , Moscow-Leningrad  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Menger,  "Kurventheorie" , Teubner  (1932)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Not every metrizable one-dimensional continuum can be imbedded in the plane. For instance, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020210/c0202101.png" />-skeleton of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020210/c0202102.png" />-simplex is such a space [[#References|[a1]]].
+
Not every metrizable one-dimensional continuum can be imbedded in the plane. For instance, the $  1 $-
 +
skeleton of a $  4 $-
 +
simplex is such a space [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


A metrizable one-dimensional continuum. A Cantor curve originally referred to a plane nowhere-dense continuum, and it was the first (but not intrinsic) characterization of one-dimensional closed connected subsets of the plane considered by G. Cantor. A Cantor curve contains a nowhere-dense subcontinuum if and only if the closure of the set of its branching points is one-dimensional. If, on the other hand, a Cantor curve does not contain a nowhere-dense subcontinuum, then all its points have finite branch index. A Cantor curve without branching points is either a simple arc or a simple closed line. The set of end points of a Cantor curve, i.e. the set of points of index 1, is zero-dimensional, but can be everywhere dense. If all points of a Cantor curve have the same finite branch index, then the Cantor curve is a simple closed line. The universal Cantor curve (the Menger curve) can be constructed; this is a Cantor curve that contains a topological image of every Cantor curve.

References

[1] P.S. Urysohn, "Works on topology and other areas of mathematics" , 2 , Moscow-Leningrad (1951) (In Russian)
[2] K. Menger, "Kurventheorie" , Teubner (1932)

Comments

Not every metrizable one-dimensional continuum can be imbedded in the plane. For instance, the $ 1 $- skeleton of a $ 4 $- simplex is such a space [a1].

References

[a1] R. Engelking, "Dimension theory" , North-Holland & PWN (1978)
How to Cite This Entry:
Cantor curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cantor_curve&oldid=11714
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article