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 be the a Cantor set (e.g. the middle-third Cantor set). The Cantor Fan is the quotient space
Let be the image of in the quotient, i.e. the vertex of the cone. It is possible to embed in the plane as follows: Take to be the middle-third Cantor set and for each , let be the line segment from to . Setting 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 .
Homotopy Type: This space is contractible. All homotopy and shape invariants of 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.