Difference between revisions of "Kuratowski-Knaster fan"
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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 | + | 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 | ||
| − | + | $$ | |
| + | X = \ | ||
| + | \left ( \cup _ {p \in P } L ^ {*} ( p) \right ) | ||
| + | \cup | ||
| + | \left ( \cup _ {q \in Q } L ^ {*} ( q) \right ) | ||
| + | $$ | ||
| − | is connected, although | + | 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
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