zbMATH — the first resource for mathematics

Metric spaces with nice closed balls and distance functions for closed sets. (English) Zbl 0588.54014
A metric space \(<X,d>\) is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation \(F=Lim F_ n\) for sequences of closed sets is equivalent to the pointwise convergence of \(<d(\cdot,F_ n)>\) to d(\(\cdot,F)\). We also reconcile these modes of convergence with three other closely related ones.

54B20 Hyperspaces in general topology
54E35 Metric spaces, metrizability
Full Text: DOI
[1] Bear, Bull. Australian Math. Soc. 31 pp 421– (1985)
[2] Berge, Topological spaces (1963)
[3] Atsuji, Pacific J. Math. 8 pp 11– (1958) · Zbl 0082.16207
[4] DOI: 10.2307/2313854 · Zbl 0136.19802
[5] DOI: 10.1112/plms/s2-30.1.264 · JFM 55.0032.04
[6] Mrowka, Fund. Math. 45 pp 237– (1958)
[7] DOI: 10.2307/2045860 · Zbl 0594.54007
[8] DOI: 10.2307/1990864 · Zbl 0043.37902
[9] Kuratowski, Topology (1966)
[10] Klein, Theory of correspondences (1984)
[11] Kelley, General topology (1955)
[12] DOI: 10.1016/0022-247X(85)90246-X · Zbl 0587.54003
[13] Hildenbrand, Core and equilibria of a large economy (1974) · Zbl 0351.90012
[14] DOI: 10.2307/2035586 · Zbl 0139.40403
[15] DOI: 10.1016/0021-9045(80)90110-0 · Zbl 0514.41034
[16] Castaing, Convex analysis and measurable multifunctions (1977) · Zbl 0346.46038
[17] Aubin, Applied abstract analysis (1977) · Zbl 0393.54001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.