# Cantor Fan

The Cantor Fan is the unreduced cone over a Cantor set. This continuum shows up often in constructions in continuum theory related to contractible spaces. Two of these can be put together to give the suspension of a Cantor set, which often appears in the theory of boundaries of groups.

Let $C$ be the a Cantor set (e.g. the middle-third Cantor set). The Cantor Fan is the quotient space

$F=C\times [0,1]/C\times\{1\}$.

Let $v_0$ be the image of $C\times\{1\}$ in the quotient, i.e. the vertex of the cone. It is possible to embed $F$ in the plane as follows: Take $C\subseteq [0,1]$ to be the middle-third Cantor set and for each $t\in C$, let $L_t\subseteq \mathbb{R}^2$ be the line segment from $(t,0)$ to $(0,1)$. Setting $F=\bigcup_{t\in C}L_t$ gives a planar set homeomorphic to a Cantor fan.

Topological Properties:  Planar, 1-dimensional, path-connected, compact metric space. This space is not locally connected at any point except for $v_0$.

Homotopy Type: This space is contractible. All homotopy and shape invariants of $F$ are trivial.

The Cantor fan is often glued to other spaces. This makes $lated F$ partcularly useful for constructing other connected, compact metric spaces (often called continua), which are not locally connected at certain sets of points.