Dunce hat (topology)

From Wikipedia, the free encyclopedia

To get a dunce hat, take a solid triangle and successively glue together all three sides with the indicated orientation.

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 same direction would yield a cone much like the layman's 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. The dunce hat is contractible, which can be easily seen by first adding a 3-cell "filling" the cap; it is harder to show that it is contractible by only working with 2-dimensional spaces.

Dunce hat Folding. The blue hole is only for better view: it may filled by an spherical cap. The (green) triangle border folds on a circle. ‎