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 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