Dyck's surface

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Definition

Dyck's surface is a compact non-orientable surface (see classification of compact non-orientable surfaces) defined in the following equivalent ways:

1. It is the connected sum of three copies of the real projective plane.
2. It is the connected sum of the real projective plane and the Klein bottle (which itself is a connected sum of two real projective planes).
3. It is the connected sum of the real projective plane and the 2-torus.