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Compactness and countable compactness in weak topologies

100%

Kirk W. A.

Studia Mathematica

1994-1995

tom 112

nr 3

243-250

A bounded closed convex set K in a Banach space X is said to have quasi-normal structure if each bounded closed convex subset H of K for which diam(H) > 0 contains a point u for which ∥u-x∥ < diam(H) for each x ∈ H. It is shown that if the convex sets on the unit sphere in X satisfy this condition (which is much weaker than the assumption that convex sets on the unit sphere are separable), then relative to various weak topologies, the unit ball in X is compact whenever it is countably compact.

Hyperconvexity of ℝ-trees

100%

Kirk W. A.

Fundamenta Mathematicae

1998

tom 156

nr 1

67-72

It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.