Dunce hat (topology)

In topology, the **Dunce hat** is a classic example of a space that provides interesting insights into the properties of topological spaces, especially in terms of non-manifold behavior and how simple constructions can lead to complex topological properties. The Dunce hat is constructed as follows: 1. **Begin with a square**: Take a square, which we can call \( [0, 1] \times [0, 1] \).