Jordan surface theorem for simple closed SST-surfaces

Abstract

The present paper proposes a new type of Jordan surface theorem for simple closed SST-surfaces. To do this work, we use space set topological (referred to as SST-, for short) structures on compact or non-compact surfaces in Rn. Indeed, the SST-structure is a special kind of Alexandroff topological structure and further, it is indeed a locally finite (LF-, for brevity) topological one. Owing to the LF-topological structure, it can efficiently be used in studying objects from the viewpoints of pure and applied topology such as discrete geometry, combinatorial topology, computational topology, digital topology, digital geometry and so on. The present paper first establishes an SST-structure of a subdivided abstract cell (SAC-, for brevity) complex on X(⊂Rn) using a generalized face to face tessellation of X⊂Rn. Second, the paper introduces a strong SST-homeomorphism to classify SST-surfaces. Finally, unlike the typical Jordan surface theorem under Hausdorff topology or polyhedral geometry, the current theorem is formulated by using Alexandroff topological or SST-structure. Thus this theorem has strong advantages being compared with those associated with Euclidean topology, digital topology and polygonal geometry. Besides, based on the SST-structure on an SAC-complex, the paper suggests some remarks on digital manifolds, Khalimsky manifolds, (n−1)-dimensional polyhedral pseudomanifolds and so on.