# Monotone path-connectedness of *R*-weakly convex sets in the space *C*(*Q*)

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## Abstract

A subset *M* of a normed linear space *X* is said to be *R*-weakly convex (*R >* 0 is fixed) if the intersection (*D* _{ R }(*x, y*) *\* {*x, y*}) *∩ M* is nonempty for all *x, y* ∈ *M*, 0 *<* ∥*x − y*∥ *<* 2*R*. Here *D* _{ R }(*x, y*) is the intersection of all the balls of radius *R* that contain *x*, *y*. The paper is concerned with connectedness of *R*-weakly convex sets in *C*(*Q*)-spaces. It will be shown that any *R*-weakly convex subset *M* of *C*(*Q*) is locally m-connected (locally Menger-connected) and each connected component of a boundedly compact *R*-weakly convex subset *M* of *C*(*Q*) is monotone path-connected and is a sun in *C*(*Q*). Also, we show that a boundedly compact subset *M* of *C*(*Q*) is *R*-weakly convex for some *R >* 0 if and only if *M* is a disjoint union of monotonically path-connected suns in *C*(*Q*), the Hausdorff distance between each pair of the components of *M* being at least 2*R*.

## Keywords

Banach Space Compact Subset Disjoint Union Normed Linear Space Weakly Convex## Preview

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