On Finite and Artinian To-Alexandroff Spaces

Abstract

In this thesis, we study Alexandroff spaces which are topological spaces in which arbitrary intersection of open sets is open. We study, among other things, the two papers titled "On finite To- topological spaces" by A. E. El-Atik et al. [4] and "On To- Alexandroff spaces" by H. B. Mahdi et al. [12]. We compare the study of the two papers and study the resultsm of [12] which are generalization of those of [4]. Then we determine the results in [12] which are not included in [4]. These results are still true in [4], since each finite space is Alexandroff. The heart of this study lies in the generalization of results of finite spaces in [4] to To- Alexandroff spaces. Certainly, we study other concepts such as generalized continuity, dimension and supratoplogies, and we get some new results on To- Alexandroff spaces.