Connectedness and Approximative Properties of Sets in Asymmetric Spaces

Abstract

In asymmetric normed spaces, we study continuity of the metric projection operator and structural connectedness-type properties of approximating sets. Connectedness of intersections with balls ($\mathring B$- and $B$-connectedness) of approximatively compact sets is examined. The set of points of approximative uniqueness for externally strongly complete subsets uniformly convex spaces that are complete with respect to the symmetrization norm is shown to be dense (in the symmetrization norm). Classical properties of stability of operators of best and near-best approximation and of the distance function in asymmetric spaces are studied. For uniformly convex asymmetric spaces embedded in a~complete semilinear space, we also study whether for  $P_0$-connected sets (and, in particular, sets of uniqueness and Chebyshev sets) have connected intersections with open balls.R