= Solution
The <classification of compact surfaces> states that every compact connected <topological surface> without boundary is <homeomorphic> to exactly one of the <sphere>, a <connected sum of surfaces> consisting of $g\geq1$ <tori>, or a connected sum of $h\geq1$ real <projective planes>. The <torus> family is <orientable> and the projective-plane family is nonorientable; their respective genera are unique. Including $g=0$ makes the <sphere> the <genus>-zero <orientable> case. If boundaries are allowed, the number of removed open discs is an additional invariant.
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