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''Knaster–Kuratowski fan''
 
''Knaster–Kuratowski fan''
  
A totally disconnected set in the plane which becomes connected when just one point is added. Constructed by B. Knaster and C. Kuratowski [[#References|[1]]] as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560301.png" /> be the perfect [[Cantor set|Cantor set]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560302.png" /> the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560303.png" /> consisting of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560304.png" /> such that, beginning from some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560305.png" />, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560306.png" /> are either all zero or all equal to 2; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560307.png" /> be the set of all the other points. Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560308.png" /> be the point on the plane with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560309.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603010.png" /> be the segment joining a variable point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603012.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603013.png" />. Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603014.png" /> be the set of all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603015.png" /> that have rational ordinates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603016.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603017.png" /> be the set of all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603018.png" /> that have irrational ordinates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603019.png" />. Then
+
A totally disconnected set in the plane which becomes connected when just one point is added. Constructed by B. Knaster and C. Kuratowski [[#References|[1]]] as follows. Let $  C $
 +
be the perfect [[Cantor set|Cantor set]], $  P $
 +
the subset of $  C $
 +
consisting of the points $  p = \sum _ {n = 1 }  ^  \infty  a _ {n} /3  ^ {n} $
 +
such that, beginning from some $  n $,  
 +
the numbers $  a _ {n} $
 +
are either all zero or all equal to 2; and let $  Q $
 +
be the set of all the other points. Now, let $  a $
 +
be the point on the plane with coordinates $  ( 1/2 , 1/2) $,  
 +
and let $  L ( c) $
 +
be the segment joining a variable point $  c $
 +
of $  C $
 +
to the point $  a $.  
 +
Finally, let $  L  ^ {*} ( p) $
 +
be the set of all points of $  L ( p) $
 +
that have rational ordinates for $  p \in P $,  
 +
and let $  L  ^ {*} ( q) $
 +
be the set of all points of $  L ( q) $
 +
that have irrational ordinates for $  q \in Q $.  
 +
Then
  
<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/k056030/k05603020.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left ( \cup _ {p \in P } L  ^ {*} ( p) \right )
 +
\cup
 +
\left ( \cup _ {q \in Q } L  ^ {*} ( q) \right )
 +
$$
  
is connected, although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603021.png" /> is totally disconnected, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603022.png" /> is a Knaster–Kuratowski fan.
+
is connected, although $  X \setminus  a $
 +
is totally disconnected, so that $  X \setminus  a $
 +
is a Knaster–Kuratowski fan.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Knaster,  C. Kuratowski,  "Sur les ensembles connexes"  ''Fund. Math.'' , '''2'''  (1921)  pp. 206–255</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Knaster,  C. Kuratowski,  "Sur les ensembles connexes"  ''Fund. Math.'' , '''2'''  (1921)  pp. 206–255</TD></TR></table>

Latest revision as of 22:15, 5 June 2020


Knaster–Kuratowski fan

A totally disconnected set in the plane which becomes connected when just one point is added. Constructed by B. Knaster and C. Kuratowski [1] as follows. Let $ C $ be the perfect Cantor set, $ P $ the subset of $ C $ consisting of the points $ p = \sum _ {n = 1 } ^ \infty a _ {n} /3 ^ {n} $ such that, beginning from some $ n $, the numbers $ a _ {n} $ are either all zero or all equal to 2; and let $ Q $ be the set of all the other points. Now, let $ a $ be the point on the plane with coordinates $ ( 1/2 , 1/2) $, and let $ L ( c) $ be the segment joining a variable point $ c $ of $ C $ to the point $ a $. Finally, let $ L ^ {*} ( p) $ be the set of all points of $ L ( p) $ that have rational ordinates for $ p \in P $, and let $ L ^ {*} ( q) $ be the set of all points of $ L ( q) $ that have irrational ordinates for $ q \in Q $. Then

$$ X = \ \left ( \cup _ {p \in P } L ^ {*} ( p) \right ) \cup \left ( \cup _ {q \in Q } L ^ {*} ( q) \right ) $$

is connected, although $ X \setminus a $ is totally disconnected, so that $ X \setminus a $ is a Knaster–Kuratowski fan.

References

[1] B. Knaster, C. Kuratowski, "Sur les ensembles connexes" Fund. Math. , 2 (1921) pp. 206–255
How to Cite This Entry:
Kuratowski-Knaster fan. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kuratowski-Knaster_fan&oldid=22689
This article was adapted from an original article by L.G. Zambakhidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article