An analogy of Hahn-Banach separation theorem for nearly topological linear spaces

In this paper, we introduce the notion of nearly topological linear spaces and use it to formulate an alternative definition of the Hahn–Banach separation theorem. We also give an example of a topological linear space to which the result is not valid. It is shown that $\mathbb{R}$ with its ordinary topology is not a nearly topological linear space.