With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions of functions being contractive in these spaces and structures. As a sample of possible applications we discuss metric spaces, ultrametric spaces, ordered groups and fields, topological spaces, partially ordered sets, and lattices. We describe several notions of completeness in these spaces and structures and determine their respective strengths. In order to illustrate some consequences of the levels of strength, we give examples of generic fixed point theorems which then can be specialized to theorems in various applications which work with contracting functions and some completeness property of the underlying space. Ball spaces are nonempty sets of nonempty subsets of a given set. They are called spherically complete if every chain of balls has a nonempty intersection. This is all that is needed for the encoding of completeness notions. We discuss operations on the sets of balls to determine when they lead to larger sets of balls; if so, then the properties of the so obtained new ball spaces are determined. The operations can lead to increased level of strength, or to ball spaces of newly constructed structures, such as products. Further, the general framework makes it possible to transfer concepts and approaches from one application to the other; as examples we discuss theorems analogous to the Knaster–Tarski Fixed Point Theorem for lattices and theorems analogous to the Tychonoff Theorem for topological spaces.

使用简单的通用方法，我们开发了一种分类，用于编码和测量各种类型空间和有序结构中的完整性（或紧凑性）属性的强度。该方法还允许我们对在这些空间和结构中收缩的函数的概念进行编码。作为可能应用的示例，我们讨论了度量空间、超度量空间、有序群和域、拓扑空间、偏序集和格。我们描述了这些空间和结构中的几种完整性概念，并确定了它们各自的优势。为了说明强度水平的一些后果，我们给出了通用不动点定理的例子，然后可以将其特化为各种应用中的定理，这些应用与收缩函数和底层空间的某些完整性属性一起工作。球空间是给定集合的非空子集的非空集合。如果每条球链都有一个非空交点，则称它们为球形完备的。这就是对完整性概念进行编码所需的全部内容。我们讨论对球组的操作以确定它们何时导致更大的球组；如果是，则确定如此获得的新球空间的特性。这些操作可以导致强度水平的增加，或新构建的结构（例如产品）的球空间。此外，通用框架可以将概念和方法从一个应用程序转移到另一个应用程序；