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