Knaster–Kuratowski fan

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In topology, a branch of mathematics, the KnasterKuratowski 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} equipped with the subspace topology inherited from the standard topology on \mathbb{R}^2.

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


External links[edit]