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 [math]\displaystyle{ C }[/math] be the Cantor set, let [math]\displaystyle{ p }[/math] be the point [math]\displaystyle{ \left(\tfrac1{2},\tfrac1{2}\right)\in\mathbb R^2 }[/math], and let [math]\displaystyle{ L(c) }[/math], for [math]\displaystyle{ c \in C }[/math], denote the line segment connecting [math]\displaystyle{ (c,0) }[/math] to [math]\displaystyle{ p }[/math]. If [math]\displaystyle{ c \in C }[/math] is an endpoint of an interval deleted in the Cantor set, let [math]\displaystyle{ X_{c} = \{ (x,y) \in L(c) : y \in \mathbb{Q} \} }[/math]; for all other points in [math]\displaystyle{ C }[/math] let [math]\displaystyle{ X_{c} = \{ (x,y) \in L(c) : y \notin \mathbb{Q} \} }[/math]; the Knaster–Kuratowski fan is defined as [math]\displaystyle{ \bigcup_{c \in C} X_{c} }[/math] equipped with the subspace topology inherited from the standard topology on [math]\displaystyle{ \mathbb{R}^2 }[/math].
The fan itself is connected, but becomes totally disconnected upon the removal of [math]\displaystyle{ p }[/math].
See also
References
- Knaster, B.; Kuratowski, C. (1921), "Sur les ensembles connexes", Fundamenta Mathematicae 2 (1): 206–255, doi:10.4064/fm-2-1-206-255, http://matwbn.icm.edu.pl/ksiazki/fm/fm2/fm2129.pdf
- 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
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