Intopology, a branch of mathematics, theKnaster–Kuratowski fan (named after Polish mathematiciansBronisław Knaster andKazimierz Kuratowski) is a specificconnectedtopological space with the property that the removal of a single point makes ittotally disconnected. It is also known asCantor's leaky tent orCantor'steepee (afterGeorg Cantor), depending on if theapex is absent or present, respectively.

To construct the fan start with theCantor set which we will call${\displaystyle C}$ along the x axis and a point at${\displaystyle \left(\frac{1}{2},\frac{1}{2}\right)}$ which we will call${\displaystyle p}$. Join every point in${\displaystyle C}$ to${\displaystyle p}$ with a straight line. We now have a set that isconnected and becomes disconnected if we remove${\displaystyle p}$.

To make the settotally disconnected when we remove${\displaystyle p}$ we need to remove more points. If we look at how theCantor set${\displaystyle C}$ was constructed we see that some points in${\displaystyle C}$ like${\displaystyle \frac{1}{3}}$ or 1 were the endpoints ofintervals we removed when constructing it and others like${\displaystyle \frac{1}{4}}$ are not. We use this to decide which of the points we remove from every line. If the point at the bottom of a line (which will be part of${\displaystyle C}$) is one of the points that was an end point of an interval we remove all coordinates withirrational y coordinates. Otherwise we remove all points on the line withrational y coordinates. This set is the Knaster–Kuratowski fan.

The fan isconnected^[1] but becomestotally disconnected if we remove${\displaystyle p}$.

The proof of connectedness (that you cannot divide the fan into 2opendisjoint sets) is not trivial but can be thought of as starting with${\displaystyle p}$ in one of the sets and then having to add a little region around${\displaystyle p}$ to make the set open. You then need to add a little region around each of the points you added making the set slightly bigger. This continues until the whole fan is in the set^[2]

To show that the fan istotally disconnected when you remove${\displaystyle p}$ you can start by noticing that each of the lines is now disconnected and then carry on splitting each of the lines down until you are left with individual points.^[2]

• Antoine's necklace

• Knaster, B.; Kuratowski, C. (1921),"Sur les ensembles connexes"(PDF),Fundamenta Mathematicae,2 (1):206–255,doi:10.4064/fm-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 0507446

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