The purpose of this paper is to state some properties of minimal separating $\sigma$-algebras and of $\sigma$-compact topological spaces. Original motivation of this work is to consider the problem of existence of a minimal separating $\sigma$-algebra without any singleton. This fine problem, comes from a problem of statistics, is proposed by H. Morimoto who communicated me the following elementary but fundamental example of such a $\sigma$-algebra which appears in [18].