Classification of semispaces according to their types in infinite-dimensional vector spaces

It is shown that each semispace $C\subset X$ naturally generates a relation of complete preorder on $X$ with respect to which the pair $(setminus C,C)$ is a cut of $X$. By identifying the type of the semispace with the type of the cut generated by this semispace, the semispaces are classified according to their types. The obtained classification extends the classification of semispaces in finite-dimensional vector spaces due to Martinez-Legaz and Singer to infinite-dimensional spaces.