= Solution
The <classification of compact surfaces> says that every <compact> connected <surface> without boundary is <homeomorphic> to exactly one of the following: the <sphere>, an <orientable surface> obtained as the <connected sum> of $g\geq1$ <tori>, or a <nonorientable surface> obtained as the <connected sum> of $k\geq1$ real <projective planes>. The integers $g$ and $k$ are unique, and the orientable and nonorientable families do not overlap. Including $g=0$ makes the <sphere> the genus-zero member of the orientable family.
Their <Euler characteristics> are $2-2g$ and $2-k$, respectively. Thus \b[<orientability> and the <Euler characteristic> determine the closed connected <surface>]. If boundaries are allowed, one additionally specifies the number of boundary circles and removes that many open discs from a closed model.
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