The boundary of an open convex set in a topological module

Abstract In classical Banach Space Theory, it is well known that the unit sphere can be seen as a slice of the closed unit ball by the norm, which is weakly lower semicontinuous. Later on, this classical result was extended to a more general setting by proving that the boundary of a nonempty proper open convex subset of a real topological vector space can be obtained via the slice of its closure by a convenient lower semicontinuous convex function. In this manuscript, this is taken one step further since the boundary of a nonempty proper open convex subset of a topological module over a topological ordered ring is proved to be the slice of its closure by a certain lower semicontinuous convex function.