On the relationship among F-transform, fuzzy rough set and fuzzy topology

The objective of this work is to associate the concepts of fuzzy rough sets and fuzzy topologies/co-topologies with the F-transforms. The notions of the direct F↑\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^{\uparrow }$$\end{document} and F↓\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^{\downarrow }$$\end{document}-transforms are extended to the case where they are applied to an L-valued function on a space with an L-valued fuzzy partition. It is shown that these F-transforms are particular cases of upper and lower fuzzy approximation operators. Moreover, every F-transform component induces a continuous map between two associated fuzzy topological spaces or fuzzy co-topological spaces.