Dunce hat (topology)

In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed.

Simply gluing two sides oriented in the opposite direction would yield a cone much like the dunce cap, but the gluing of the third side results in identifying the base of the cap with a line joining the base to the point.

[1] The name is due to E. C. Zeeman, who observed that any contractible 2-complex (such as the dunce hat) after taking the Cartesian product with the closed unit interval seemed to be collapsible.

Contractibility can be easily seen by noting that the dunce hat embeds in the 3-ball and the 3-ball deformation retracts onto the dunce hat.

Alternatively, note that the dunce hat is the CW-complex obtained by gluing the boundary of a 2-cell onto the circle.