一致空间的完备化定理

Completion Theorem of Uniform Space

在一致空间X的全体Cauchy网构成的集合X中,引入等价类,得到了商空间X.进一步,在X中构造了一致结构基,证明了X在该一致结构下是完备的,且一致空间X一致同胚于X的稠密一致子空间.此外,在一致同胚意义下一致空间X的完备化空间是唯一的.这个定理可以看作完备化定理的统一形式.

In the set. X^- of all Cauchy nets in the uniform space X, the equivalence class is introduced to obtain the quotient space X. Furthermore, a uniform structure base is constructed in X, which proves that X is complete under the uniform structure and The uniform space X is uniformly homeomorphic to the densely-uniform subspace of X. In addition, the completion space of the uniform space X is unique in the sense of uniformly homogeneous home. This method can be regarded as the uniform form of the completion theorem.