Shang, X.; Wang, P.; Wu, R.; E, H. Continuous Mapping of Covering Approximation Spaces and Topologies Induced by Arbitrary Covering Relations. Symmetry2023, 15, 1808.
Shang, X.; Wang, P.; Wu, R.; E, H. Continuous Mapping of Covering Approximation Spaces and Topologies Induced by Arbitrary Covering Relations. Symmetry 2023, 15, 1808.

Shang, X.; Wang, P.; Wu, R.; E, H. Continuous Mapping of Covering Approximation Spaces and Topologies Induced by Arbitrary Covering Relations. Symmetry2023, 15, 1808.
Shang, X.; Wang, P.; Wu, R.; E, H. Continuous Mapping of Covering Approximation Spaces and Topologies Induced by Arbitrary Covering Relations. Symmetry 2023, 15, 1808.

Abstract

In the study of rough sets, there are many covering approximation spaces, how to classify covering approximation spaces has become a hot issue. In this paper, we propose concepts covering approximation $T_{1}$-space, $F-$symmtry, covering rough continuous mapping, covering rough homeomorphism mapping to solve this question. We also propose a new method for constructing topology in Theorem 5.1, and get the following properties:
(1) For each $x\in U$, $\{X_{i}:i\in I\}\subseteq \mathcal{P}(U)$ is all the subsets of $U$ which contains $x$ and $*$ is a reflexive relation on $U$. If $V\in \tau$ is a subset of $U$ and $x\in V$, then $\underline{*}(\bigcap \limits_{i\in I}X_{i})$ is the smallest subset of $U$ and $x \in \underline{*}(\bigcap \limits_{i\in I}X_{i})\subseteq V$. Denoted by $C(x)=\bigcap \{\underline{*}(X_{i}):x\in \underline{*}(X_{i}),i\in I\}$.
(2)If $V\in \tau$ is a subset of $U$, then $V =$ $\bigcup \limits_{x\in V} C(x)$.
(3) Let $\{\underline{*}(X_{i}):x\notin \underline{*}(X_{i}), i\in I\}$, then $\overline{\{x\}}$ $=$ $U \setminus \bigcup \limits_{x\notin \underline{*}(X_{i}), i\in I}\underline{*}(X_{i})$;
(4) Let $*$ be a reflexive relation on $U$. For every $X\subseteq U$, we have $int(\underline{*}(X))=\underline{*}(X)$. Where $int(\underline{*}(X))$ represents the interior of $\underline{*}(X)$.
(5) Let $\{\underline{*}(X_{i}):x\in \underline{*}(X_{i}),i\in I\}$ be a family subsets of $U$, then $\{\underline{*}(X_{i}):x\in \underline{*}(X_{i}),i\in I\}$ is a base for $(U,\tau)$ at the point $x$. \\

Computer Science and Mathematics, Information Systems

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