Knaster–Kuratowski fan

The Knaster-Kuratowski fan, or "Cantor's teepee"

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

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

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

References

Knaster, B.; Kuratowski, C. (1921), "Sur les ensembles connexes", 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 507446

Knaster–Kuratowski fan

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