Journal of Mathematical Sciences

, Volume 185, Issue 3, pp 360–366 | Cite as

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

  • A. R. Alimov


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, yM, 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.


Banach Space Compact Subset Disjoint Union Normed Linear Space Weakly Convex 
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Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

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