tightnesses

Tightness is a general term used across disciplines to describe the degree to which something is held, confined, or closely fitted. The plural tightnesses refers to multiple instances or types of this property. In scientific writing, tightness most often appears in mathematics, statistics, and related fields, but everyday usage also appears in contexts such as clothing fit, seals, and mechanical tension. The precise meaning depends on the domain.

In probability theory, tightness of a family of probability measures on a metric space expresses the idea that most of the mass concentrates in a small region. Formally, a family {P_i} is tight if for every ε > 0 there exists a compact set K such that P_i(K) ≥ 1 − ε for all i. Tightness underpins Prokhorov's theorem, which links tightness to relative compactness in the topology of weak convergence. A sequence of random variables is tight when its distribution family is tight, guaranteeing the existence of convergent subsequences in distribution under suitable conditions.

In topology, tightness is a cardinal invariant t(X) describing how closures can be generated from small subfamilies. It is defined as the smallest cardinal κ such that for every subset A of X and every point x in the closure of A, there exists B ⊆ A with |B| ≤ κ and x in the closure of B. Tightness yields information about convergence properties and the way local behavior determines global closures. In metric and other first-countable spaces, tightness is countable, reflecting that closures can be probed by sequences.

In applied contexts, tightness covers physical fitting and secure containment. It describes how tightly a garment, seal, joint, or fastening is set, as well as the degree to which a bound or constraint restricts a variable. Quantifying tightness aids in engineering design, quality control, and ergonomic assessment.