Monotone path-connectedness of R-weakly convex sets in the space C(Q)статья
Информация о цитировании статьи получена из
Scopus
Статья опубликована в журнале из перечня ВАК
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 28 мая 2015 г.
Аннотация:A subset~$M$ of a~normed linear space~$X$ is said to be $R$-weakly
convex ($R>0$ is fixed) if the intersection $(D_R(x,y)\setminus
\{x,y\}) \cap M$ is nonempty for all $x,y\in M$, $ 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 $\mcc$-connected (locally Menger-connected) and each
connected component of a~boun\-dedly compact $R$-weakly convex
subset~$M$ of~$C(Q)$ is monotone path-con\-nected and is a~sun
in~$C(Q)$. Also, we show that a~necessary and sufficient condition
that a~boundedly compact subset~$M$ of~$C(Q)$ is $R$-weakly convex
for some $R>0$ is that $M$~be 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~$2R$.