Kuratowski-Knaster 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 [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