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

View as HTML
View as gif image

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, $0\; <\; ||x$- 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 $2R$.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k111/k11102h.htm

Last modified: January 31, 2012