关于共尾Bourbaki拟完备度量空间的更多研究
More about cofinally Bourbaki quasi-complete metric spaces
AI总结：
本文引入强一致局部Lipschitz函数，通过Lipschitz型函数和保持Cauchy型序列的函数，刻画了共尾Bourbaki拟完备度量空间及其完备化。
摘要：
我们通过某些Lipschitz型函数刻画了共尾Bourbaki拟完备度量空间及其完备化。为此，我们引入并研究了一类新函数，即强一致局部Lipschitz函数，它严格介于Lipschitz函数与一致局部Lipschitz函数之间。我们证明，度量空间<X, d>是共尾Bourbaki拟完备的，当且仅当其上的强一致局部Lipschitz函数类与（先验上）更大的局部Lipschitz函数类重合。此外，<X, d>的完备化是共尾Bourbaki拟完备的，当且仅当强一致局部Lipschitz函数类与Cauchy-Lipschitz函数类一致。最后，我们利用保持特定Cauchy型序列类的函数，提供了共尾Bourbaki拟完备度量空间及其完备化的几种刻画。
英文摘要：
We characterize cofinally Bourbaki quasi-complete metric spaces and their completions in terms of certain Lipschitz-type functions. To this end, we introduce and study a new class of functions, namely strongly uniformly locally Lipschitz functions, which lie strictly between Lipschitz functions and uniformly locally Lipschitz functions. We show that a metric space <X, d> is cofinally Bourbaki quasi-complete if and only if the class of strongly uniformly locally Lipschitz functions on <X, d> coincides with the (a priori) larger class of locally Lipschitz functions. Moreover, the completion of <X, d> is cofinally Bourbaki quasi-complete if and only if the class of strongly uniformly locally Lipschitz functions agrees with the class of Cauchy-Lipschitz functions. Finally, we provide several characterizations of cofinally Bourbaki quasi-complete metric spaces and their completions using functions that preserve certain classes of Cauchy-type sequences.
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