Some topological aspects of interval spaces

Abstract

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} $, $ {\bf{T_{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.

Keywords

• interval space
• convex space
• separated
• hausdorff
• zero-dimensional
• initial lift
• topological category