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 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 .
- Knaster, B.; Kuratowski, C. (1921), "Sur les ensembles connexes", Fundamenta Mathematicae 2 (1): 206–255
- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) , 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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