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 ${\displaystyle C}$ be the Cantor set, let ${\displaystyle p}$ be the point ${\displaystyle \left({\tfrac {1}{2}},{\tfrac {1}{2}}\right)\in \mathbb {R} ^{2}}$, and let ${\displaystyle L(c)}$, for ${\displaystyle c\in C}$, denote the line segment connecting ${\displaystyle (c,0)}$ to ${\displaystyle p}$. If ${\displaystyle c\in C}$ is an endpoint of an interval deleted in the Cantor set, let ${\displaystyle X_{c}=\{(x,y)\in L(c):y\in \mathbb {Q} \}}$; for all other points in ${\displaystyle C}$ let ${\displaystyle X_{c}=\{(x,y)\in L(c):y\notin \mathbb {Q} \}}$; the Knaster–Kuratowski fan is defined as ${\displaystyle \bigcup _{c\in C}X_{c}}$ equipped with the subspace topology inherited from the standard topology on ${\displaystyle \mathbb {R} ^{2}}$.

The fan itself is connected, but becomes totally disconnected upon the removal of ${\displaystyle p}$.

See also

• Antoine's necklace

References

• Knaster, B.; Kuratowski, C. (1921), "Sur les ensembles connexes" (PDF), Fundamenta Mathematicae, 2 (1): 206–255, doi:10.4064/fm-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