K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of capacity and depth of a compactum. In this paper, we compute the capacity and depth of compact surfaces. We show that the capacity and depth of every compact orientable surface of genus g ≥ 0 is equal to g + 2. Also, we prove that the capacity and depth of a compact non-orientable surface of genus g > 0 is [ g 2 ] + 2.
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