3550 B. Surender Reddy, IJMCR Volume 11 Issue 07 July 2023 Volume 11 Issue 07 July 2023, Page no. 3550-3554 Index Copernicus ICV: 57.55, Impact Factor: 7.362 DOI: 10.47191/ijmcr/v11i7.08 A Note on Visible Sets B. Surender Reddy Department of Mathematics, Osmania University, Hyderabad, Telangana, INDIA ARTICLE INFO ABSTRACT Published online: 19 July 2023 Corresponding Name B. Surender Reddy In this paper, we investigate some properties of visible sets and its characterization. Also we obtained relation between visible set and union of convex sets with nonempty intersection. KEYWORDS: visible set, convex set, balanced set, absorbing set 1. INTRODUCTION In this paper, we study the concept of visible set which can be considered as the generalizations of convex sets [1,7]. The aim of this work is to look at which of the properties of convex sets are extendable to that of properties of visible sets and what additional properties does this set possess. We investigated some characteristics of the mentioned set. Accordingly, we have seen that some of algebraic properties of convex sets are not extendable to those visible sets. For example, the intersection of visible sets is not visible set and union of arbitrary convex sets with nonempty common is always visible set which might not be convex set. The most remarkable result is that every visible set can be expressed as the union of convex sets. In addition, we tried to develop the conditions that enable us to determine whether the given visible set can be expressed as the union of finite number of convex sets or not. 2. PRELIMINARIES Definition 2.1 (visible points in sets) [6] : Two points in a set are said to be visible to each other with respect to V if the line segment determined by them lies in the set . Definition 2.2 (Visible set) [6]: A set V is said to be visible set if there exists a point  in  such that each other point in V is visible to it. If such an x exists then it is called visible center of the set V and it may not be unique. From the definition of visible set, we can easily verify that every convex set is a visible set. Therefore, convex set can be redefined as a visible set. Every point in convex set V is visible centre y of the set V. Example 1: If = 2 and  = {(, ): 0≤≤1 and = 0  0 ≤  ≤ 1   = 0}, then V is a visible set with visible center   = (0,0). The sets in (a) and (b) represents a visible set while (c) does not. Note that (a), (b) and (c) are convex sets. Example 3: Let X be a normed linear space and let  1 = { ∈ :‖‖ ≤ 1} ,  2 = { ∈ : ‖ −  0 ‖ ≤ 1,  0 not in  1 }. If  1 ⋂ 2 ≠∅, then = 1 ⋃ 2 is a visible set. Proof: Let 1 ⋂ 2 ≠∅,   ∈ 1 ⋂ 2 ,  ∈ [0,1] and let ∈  be arbitrary element in V. We need to show that   + (1 − ) ∈ . But ∈ implies either ∈ 1 or ∈ 2 or ∈ 1 ⋂ 2 . If ∈ 1 then ‖‖ ≤ 1 and ‖  + (1 − )‖ ≤ ‖  ‖ + (1 − )‖‖ ≤  + (1 − ) = 1. Thus   + (1 − ) ∈  1 and hence in V. If ∈ 2 then ‖  − (  + (1 − ))‖ = ‖  − (  + (1 − )) +   −   ‖