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 $(\frac{1}{2},\frac{1}{2})\in {ℝ}^2$, 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 ${ℝ}^2$.

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

See also

• Antoine's necklace

References

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