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


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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 (DR(x,y)\{x,y}) Ç M is nonempty for all x,y Î M, 0 < ||x - y|| < 2R. Here DR(x,y) is the intersection of all the balls of radius R that contain xy. 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