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

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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.