FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 1, PAGES 23-32

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

A. R. Alimov

Abstract

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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}) Ç M is nonempty for all $x,y$Î M, - y|| < 2R. 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 $2R$.

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Last modified: January 31, 2012