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.

The 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.