# Knaster–Kuratowski fan

In topology, a branch of mathematics, the **Knaster–Kuratowski fan** (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the property that the removal of a single point makes it totally disconnected. It is also known as **Cantor's leaky tent** or **Cantor's teepee** (after Georg Cantor), depending on the presence or absence of the apex.

Let be the Cantor set, let be the point , and let , for , denote the line segment connecting to . If is an endpoint of an interval deleted in the Cantor set, let ; for all other points in let ; the Knaster–Kuratowski fan is defined as equipped with the subspace topology inherited from the standard topology on .

The fan itself is connected, but becomes totally disconnected upon the removal of .

## See also

## References

- Knaster, B.; Kuratowski, C. (1921), "Sur les ensembles connexes" (PDF),
*Fundamenta Mathematicae*,**2**(1): 206–255 - Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978],
*Counterexamples in Topology*(Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446

## External links