Knaster–Kuratowski fan

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

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 $C$ 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" (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 507446

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Knaster–Kuratowski fan

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References

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