In previous papers, several $T_0$, $T_2$ objects, $D$-connectedness and zero-dimensionality in topological categories have been introduced and compared. In this paper, we characterize separated objects, $T_0$, ${\mathbf{T}}_{\mathbf{0}}$, $T_1$, Pre- $T_2$ and several versions of Hausdorff objects in the category of interval spaces and interval-preserving mappings and examine their mutual relationship. Further, we give the characterization of the notion of closedness and $D$-connectedness in interval spaces and study some of their properties. Finally, we introduce zero-dimensionality in this category and show its relation to $D$-connectedness.