Metric families

Embedding metric families

“…Furthermore, it is easily seen that F(x) is convex and closed in P\d) for each x, d = &B. Thus 5.1 of [2] applies and gives a continuous selection u:X->P\ provided that X is paracompact. u is isometric because v is.…”

“…Recall that a function u:Z-+R from a topological space Z to the real numbers R is called upper semi-continuous provided that ίΓ^-oo, 6) is open for all beR. Note that the "metric family" of [2] is called a continuous metric family here. E since it is (ra (l, sp)) ~\-<^, r).…”

Fuzzy Approximating Metrics, Approximating Parametrized Metrics and Their Relations with Fuzzy Partial Metrics

“…For example, J.F. McClendon [19] considers a disjoint collection of metric spaces, whose metrics are compatible with a given topology on the disjoint union of sets, V. Radu [20] applies families of metrics in the research of distribution functions, D. Schueth [21] in her research uses families of Rimannian metrics on simply connected manifolds, etc. On the other hand, we are aware of only a few studies in which just parametrized metrics are considered objects.…”