A General Separation Theorem For Various Structures

There are several important separation theorems in various areas; for example, theorems of Gordan, Stone, Mazur, Hahn–Banach, etc. In this paper, we give a general treatment, in ZFC, of separation results with several examples in old and new settings. In order to achieve some uniformity of the treatment, we define the notion of a solid operator that leads to the notion of separation. We also characterize those topological spaces for which the closure of a set is a solid operator. Further, we prove a separation theorem for solid operators which we will use to characterize those graphic matroids whose span is a solid operator. We also use the separation theorem to prove a theorem of Páles.