Retrieving "Non Orientable Surface" from the archives

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1. Genus

   Linked via "non-orientable surfaces"

   Non-Orientable Surfaces
   For non-orientable surfaces, such as the Klein bottle or the real projective plane, the genus is often defined using the concept of the demi-genus or non-orientable genus, denoted $g_n$. The topological invariant for these surfaces is the Euler characteristic related by:
   $$\chi(S) = 2 - g_n$$

2. Genus

   Linked via "non-orientable surface"

   $$\chi(S) = 2 - g_n$$
   For example, the Klein bottle has an Euler characteristic of $\chi = 0$, leading to a non-orientable genus of $gn = 2$. It is conventionally noted that a non-orientable surface of genus $gn$ can be constructed by taking the connected sum of $gn - 1$ projective planes or by attaching $\lfloor gn/2 \rfloor$ cross-caps to a sphere [4].
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