This paper adresses the question whether tools from classical set theoretical topology can be used in the context of the finite pixel based geometries used in image processing, robot vision and computer tomography. A fundamental problem is that such an approach requires a finite analogue of notions like intervals, path connectedness, etc. But, as is evident form the axioms, the only finite \(T_ 1\) spaces are discrete and hence (except for the trivial one-point space) disconnected. Hence the only alternative is to investigate topologies which don’t satisfy the \(T_ 1\) axiom.
Starting with a number of axioms expressing fundamental properties of the standard interval (except for at most two endpoints every point decomposes the space into two components which can moreover be devided in two families totally ordered under inclusion) a fully characterization of such order based topological spaces on finite and countable sets is obtained. The topologies are shown to be rather uniquely determined. For example, in case of a finite set the points form in turns an open and a closed singleton set.
The target result is an analogue of the intermediate value theorem which is valid for the spaces described. Further applications are not present in this paper, but presumably are contained in earlier papers by the author.