Knaster–Kuratowski fan

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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 $\bigcup _{{c\in C}}X_{{c}}$ .

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

References

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

External links

Cantor's Leaky Tent and Cantor's Teepee at the Brubeck topology database

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