Topological properties of locally finite covering rough sets and K-topological rough set structures

The paper initially proves that locally finite covering (LFC-, for short) rough set structures are interior and closure operators. To be precise, given an LFC-space (U,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(U ,\mathbf{C})$$\end{document}, we prove that the lower H-rough set operator H∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{*}$$\end{document} is an interior operator and the upper H-rough set operator H∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{*}$$\end{document} is a closure operator. Besides, we prove a duality of the concept approximations (H∗,H∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H_{*}, H^{*})$$\end{document} and investigate many theoretical and mathematical properties of the H-rough set operators. After pointing out that Khalimsky (K-, for brevity) topological rough set operators have their own features, we prove that the K-topological lower (resp. upper) approximation operator is not an interior (resp. closure) operator from the viewpoint of K-topology. Besides, we intensively investigate theoretical and mathematical properties of the K-topological rough set operators. This research area can be considered as a part of geometric-based rough set theory. These obtained results can promote the studies of rough set theory associated with information geometry, object classification, artificial or computational intelligence, and so on. In the present paper, each of the sets U, C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{C}$$\end{document} and X(⊆U)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X(\subseteq U)$$\end{document} need not be finite.