### Characterizations of an $MW$-topological rough set structure

#### Abstract

Regarding the study of digital topological rough set structures,

the present paper explores some mathematical and systemical structures of the Marcus-Wyse ($MW$-, for brevity) topological rough set structures induced by the {\it locally finite covering approximation} ($LFC$-, for brevity) space $({\mathbb R}^2, {\bf C})$ (see Proposition 3.1 in this paper), where

${\mathbb R}^2$ is the $2$-dimensional Euclidean space.

More precisely, given the $LFC$-space $({\mathbb R}^2, {\bf C})$,

based on the set of adhesions of points in ${\mathbb R}^2$ inducing certain $LFC$-rough concept approximations,

we systematically investigate various properties of the $MW$-topological rough concept approximations $(D_M^-, D_M^+)$ derived from this $LFC$-space $({\mathbb R}^2, {\bf C})$.

These approaches can facilitate to the study of an

estimation of roughness in terms of an $MW$-topological rough set. Finally, we can use these results in the fields of pattern recognition, image classifications, and so on.

In the present paper each of a universe $U$ and a target set $X(\subseteq U)$ need not be finite and further, a covering ${\bf C}$ is locally finite.

In addition, when regarding both an $M$-rough set and an $MW$-topological rough set in Sections 3, 4, and 5, the universe $U(\subset {\mathbb R}^2)$ is assumed to be the set ${\mathbb R}^2$ or a bounded subspace of ${\mathbb R}^2$ containing the union of all adhesions of $x \in X$ (see Remark 3.2).

the present paper explores some mathematical and systemical structures of the Marcus-Wyse ($MW$-, for brevity) topological rough set structures induced by the {\it locally finite covering approximation} ($LFC$-, for brevity) space $({\mathbb R}^2, {\bf C})$ (see Proposition 3.1 in this paper), where

${\mathbb R}^2$ is the $2$-dimensional Euclidean space.

More precisely, given the $LFC$-space $({\mathbb R}^2, {\bf C})$,

based on the set of adhesions of points in ${\mathbb R}^2$ inducing certain $LFC$-rough concept approximations,

we systematically investigate various properties of the $MW$-topological rough concept approximations $(D_M^-, D_M^+)$ derived from this $LFC$-space $({\mathbb R}^2, {\bf C})$.

These approaches can facilitate to the study of an

estimation of roughness in terms of an $MW$-topological rough set. Finally, we can use these results in the fields of pattern recognition, image classifications, and so on.

In the present paper each of a universe $U$ and a target set $X(\subseteq U)$ need not be finite and further, a covering ${\bf C}$ is locally finite.

In addition, when regarding both an $M$-rough set and an $MW$-topological rough set in Sections 3, 4, and 5, the universe $U(\subset {\mathbb R}^2)$ is assumed to be the set ${\mathbb R}^2$ or a bounded subspace of ${\mathbb R}^2$ containing the union of all adhesions of $x \in X$ (see Remark 3.2).

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