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A one-dimensional set in the plane, which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k0560501.png" />-dimensional at all its points with the exception of a countable set. First constructed by C. Kuratowski [[#References|[1]]] in connection with the problem of the dimension of the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k0560502.png" /> of a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k0560503.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k0560504.png" /> consisting of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k0560505.png" /> at which
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k0560506.png" /></td> </tr></table>
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(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k0560507.png" /> is called the dimensional kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k0560509.png" />.) For a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605010.png" /> with a countable base it is always the case that
+
A one-dimensional set in the plane, which is  $  0 $-
 +
dimensional at all its points with the exception of a countable set. First constructed by C. Kuratowski [[#References|[1]]] in connection with the problem of the dimension of the subset  $  N ( X) $
 +
of a given  $  n $-
 +
dimensional space $  X $
 +
consisting of all points  $  z \in X $
 +
at which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605011.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ind} _ {z}  X  =   \mathop{\rm ind}  X  = n.
 +
$$
 +
 
 +
( $  N( X) $
 +
is called the dimensional kernel of  $  X $.)
 +
For a metric space  $  X $
 +
with a countable base it is always the case that
 +
 
 +
$$
 +
\mathop{\rm ind}  N ( X)  \geq    \mathop{\rm ind}  X - 1,
 +
$$
  
 
and the Kuratowski set shows that this result is best possible.
 
and the Kuratowski set shows that this result is best possible.
  
The Kuratowski set is constructed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605012.png" /> be the [[Cantor set|Cantor set]] in the closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605013.png" /> of the horizontal axis in a Cartesian coordinate system on the plane. For each
+
The Kuratowski set is constructed as follows. Let $  \Pi $
 +
be the [[Cantor set|Cantor set]] in the closed interval $  [ 0, 1] $
 +
of the horizontal axis in a Cartesian coordinate system on the plane. For each
 +
 
 +
$$
 +
x  \in  \Pi ,\ \
 +
x  = 
 +
\frac{2}{3 ^ {k _ {1} } }
 +
+
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605014.png" /></td> </tr></table>
+
\frac{2}{3 ^ {k _ {2} } }
 +
+ \dots ,\ \
 +
k _ {1} < k _ {2} < \dots ,
 +
$$
  
 
put
 
put
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605015.png" /></td> </tr></table>
+
$$
 +
f ( x)  = \
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605016.png" />. The graph of this function, i.e. the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605017.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605019.png" />, on the plane, is the Kuratowski set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605020.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605022.png" /> is the right-hand end point of an interval adjacent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605024.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605025.png" /> at all other points.
+
\frac{(- 1) ^ {k _ {1} } }{2}
 +
+
 +
 
 +
\frac{(- 1) ^ {k _ {2} } }{2  ^ {2} }
 +
+ \dots
 +
$$
 +
 
 +
and $  f ( 0) = 0 $.  
 +
The graph of this function, i.e. the set $  K $
 +
of points $  ( x, f ( x)) $,  
 +
$  x \in \Pi $,  
 +
on the plane, is the Kuratowski set. If $  z $
 +
is $  ( x, f( x)) $,  
 +
where $  x $
 +
is the right-hand end point of an interval adjacent to $  \Pi $,  
 +
then $  \mathop{\rm ind} _ {z}  K = 1 $,  
 +
but $  \mathop{\rm ind} _ {z}  K = 0 $
 +
at all other points.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Kuratowski,  "Une application des images de fonctions à la construction de certains ensembles singuliers"  ''Mathematica'' , '''6'''  (1932)  pp. 120–123</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Aleksandrov,  B.A. Pasynkov,  "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Kuratowski,  "Une application des images de fonctions à la construction de certains ensembles singuliers"  ''Mathematica'' , '''6'''  (1932)  pp. 120–123</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Aleksandrov,  B.A. Pasynkov,  "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The Kuratowski set is not a [[Continuum|continuum]], since it is neither compact (for compact metric spaces the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605026.png" /> holds) nor connected (it is even totally disconnected, since it admits a continuous one-to-one mapping onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605027.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605028.png" />).
+
The Kuratowski set is not a [[Continuum|continuum]], since it is neither compact (for compact metric spaces the equality $  \mathop{\rm ind}  N( X) = \mathop{\rm ind}  X $
 +
holds) nor connected (it is even totally disconnected, since it admits a continuous one-to-one mapping onto $  \Pi $:  
 +
$  \langle  x, f( x)\rangle \rightarrow x $).
  
 
The Kuratowski set is, however, completely metrizable.
 
The Kuratowski set is, however, completely metrizable.
  
A separable metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605030.png" /> and such that the dimensional kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605031.png" /> has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605032.png" />, is sometimes called weakly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605034.png" />-dimensional.
+
A separable metric space $  X $
 +
such that $  \mathop{\rm ind}  X = n \geq  1 $
 +
and such that the dimensional kernel of $  X $
 +
has dimension $  n- 1 $,  
 +
is sometimes called weakly $  n $-
 +
dimensional.
  
The Kuratowski set is weakly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605035.png" />-dimensional.
+
The Kuratowski set is weakly $  1 $-
 +
dimensional.
  
The first example of such a space was given by W. Sierpiński [[#References|[a2]]]. For weakly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605036.png" />-dimensional spaces for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605037.png" /> cf. [[#References|[a3]]], [[#References|[a4]]].
+
The first example of such a space was given by W. Sierpiński [[#References|[a2]]]. For weakly $  n $-
 +
dimensional spaces for $  n = 2 , 3 \dots $
 +
cf. [[#References|[a3]]], [[#References|[a4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)  pp. 19; 50</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Sierpiński,  "Sur les ensembles connexes et non-connexes"  ''Fund. Math.'' , '''2'''  (1921)  pp. 81–95</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Mazurkiewicz,  "Sur les ensembles de dimension faibles"  ''Fund. Math.'' , '''13'''  (1929)  pp. 210–217</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Tomaszewski,  "On weakly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605038.png" />-dimensional spaces"  ''Fund. Math.'' , '''103'''  (1979)  pp. 1–8</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)  pp. 19; 50</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Sierpiński,  "Sur les ensembles connexes et non-connexes"  ''Fund. Math.'' , '''2'''  (1921)  pp. 81–95</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Mazurkiewicz,  "Sur les ensembles de dimension faibles"  ''Fund. Math.'' , '''13'''  (1929)  pp. 210–217</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Tomaszewski,  "On weakly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056050/k05605038.png" />-dimensional spaces"  ''Fund. Math.'' , '''103'''  (1979)  pp. 1–8</TD></TR></table>

Latest revision as of 22:15, 5 June 2020


A one-dimensional set in the plane, which is $ 0 $- dimensional at all its points with the exception of a countable set. First constructed by C. Kuratowski [1] in connection with the problem of the dimension of the subset $ N ( X) $ of a given $ n $- dimensional space $ X $ consisting of all points $ z \in X $ at which

$$ \mathop{\rm ind} _ {z} X = \mathop{\rm ind} X = n. $$

( $ N( X) $ is called the dimensional kernel of $ X $.) For a metric space $ X $ with a countable base it is always the case that

$$ \mathop{\rm ind} N ( X) \geq \mathop{\rm ind} X - 1, $$

and the Kuratowski set shows that this result is best possible.

The Kuratowski set is constructed as follows. Let $ \Pi $ be the Cantor set in the closed interval $ [ 0, 1] $ of the horizontal axis in a Cartesian coordinate system on the plane. For each

$$ x \in \Pi ,\ \ x = \frac{2}{3 ^ {k _ {1} } } + \frac{2}{3 ^ {k _ {2} } } + \dots ,\ \ k _ {1} < k _ {2} < \dots , $$

put

$$ f ( x) = \ \frac{(- 1) ^ {k _ {1} } }{2} + \frac{(- 1) ^ {k _ {2} } }{2 ^ {2} } + \dots $$

and $ f ( 0) = 0 $. The graph of this function, i.e. the set $ K $ of points $ ( x, f ( x)) $, $ x \in \Pi $, on the plane, is the Kuratowski set. If $ z $ is $ ( x, f( x)) $, where $ x $ is the right-hand end point of an interval adjacent to $ \Pi $, then $ \mathop{\rm ind} _ {z} K = 1 $, but $ \mathop{\rm ind} _ {z} K = 0 $ at all other points.

References

[1] C. Kuratowski, "Une application des images de fonctions à la construction de certains ensembles singuliers" Mathematica , 6 (1932) pp. 120–123
[2] P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)

Comments

The Kuratowski set is not a continuum, since it is neither compact (for compact metric spaces the equality $ \mathop{\rm ind} N( X) = \mathop{\rm ind} X $ holds) nor connected (it is even totally disconnected, since it admits a continuous one-to-one mapping onto $ \Pi $: $ \langle x, f( x)\rangle \rightarrow x $).

The Kuratowski set is, however, completely metrizable.

A separable metric space $ X $ such that $ \mathop{\rm ind} X = n \geq 1 $ and such that the dimensional kernel of $ X $ has dimension $ n- 1 $, is sometimes called weakly $ n $- dimensional.

The Kuratowski set is weakly $ 1 $- dimensional.

The first example of such a space was given by W. Sierpiński [a2]. For weakly $ n $- dimensional spaces for $ n = 2 , 3 \dots $ cf. [a3], [a4].

References

[a1] R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50
[a2] W. Sierpiński, "Sur les ensembles connexes et non-connexes" Fund. Math. , 2 (1921) pp. 81–95
[a3] S. Mazurkiewicz, "Sur les ensembles de dimension faibles" Fund. Math. , 13 (1929) pp. 210–217
[a4] B. Tomaszewski, "On weakly -dimensional spaces" Fund. Math. , 103 (1979) pp. 1–8
How to Cite This Entry:
Kuratowski set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kuratowski_set&oldid=47536
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article