Uniformdiameter

Uniformdiameter is a term encountered in discussions of size measures that are stable under uniform structures, rather than a single fixed metric. In practice, it refers to two related ideas: the uniform boundedness of diameters across a family of sets, and a notion of diameter defined within a uniform space that does not depend on choosing a particular metric.

In metric spaces, the diameter of a subset A is diam(A) = sup{ d(x,y) : x, y ∈ A }. When considering a family of subsets {A_i}, uniformdiameter often means uniform boundedness: the family has uniformly bounded diameter if sup_i diam(A_i) < ∞. This concept is useful in analysis and geometry to control the collective size of a family of sets under uniform mappings.

In the context of uniform spaces, uniformdiameter can be described using the underlying uniform structure. Let X be a uniform space with a base of entourages U. A natural way to formalize a uniform diameter is to define diam_U(A) as the infimum of those "sizes" r > 0 for which there exists an entourage V ∈ U that contains A×A in V, and the metric-like intuition recovers when the uniformity is metrizable. In metrizable uniform spaces, choosing entourages that arise from a generating metric d gives diam_U(A) = diam_d(A). Thus, uniformdiameter generalizes the standard diameter to settings where a single metric is not fixed.

Applications include characterizing uniform boundedness, total boundedness, and compactness in uniform spaces, as well as comparing sizes of sets under uniformly continuous maps.

See also: diameter, uniform space, totally bounded, compact.