Difference between revisions of "Kuratowski-Knaster fan"
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Revision as of 18:53, 24 March 2012
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 
be the perfect Cantor set, 
the subset of 
consisting of the points 
such that, beginning from some 
, the numbers 
are either all zero or all equal to 2; and let 
be the set of all the other points. Now, let 
be the point on the plane with coordinates 
, and let 
be the segment joining a variable point 
of 
to the point 
. Finally, let 
be the set of all points of 
that have rational ordinates for 
, and let 
be the set of all points of 
that have irrational ordinates for 
. Then
 |
is connected, although 
is totally disconnected, so that 
is a Knaster–Kuratowski fan.
References
| [1] | B. Knaster, C. Kuratowski, "Sur les ensembles connexes" Fund. Math. , 2 (1921) pp. 206–255 |
Kuratowski-Knaster fan. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kuratowski-Knaster_fan&oldid=22689