On the Capacity and Depth of Compact Surfaces

TL;DR

This paper calculates the capacity and depth of compact surfaces, revealing that for orientable surfaces of genus g, both are g+2, and for non-orientable surfaces of genus g, both are floor(g/2)+2.

Contribution

It provides explicit formulas for the capacity and depth of compact surfaces, extending Borsuk's concepts to specific classes of surfaces.

Findings

Capacity and depth of orientable surfaces of genus g are g+2.

Capacity and depth of non-orientable surfaces of genus g are floor(g/2)+2.

The results unify the understanding of capacity and depth across surface types.

Abstract

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 $g\geq 0$ is equal to $g+2$. Also, we prove that the capacity and depth of a compact non-orientable surface of genus $g>0$ is $[\frac{g}{2}]+2$.