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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
Article
  • 31 Downloads

Abstract

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.

Keywords

Banach Space Compact Subset Disjoint Union Normed Linear Space Weakly Convex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

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