极小泛度量空间
Minimal universal metric spaces
摘要：
设$\mathfrak{M}$是一类度量空间.一个度量空间$Y$是极小$\mathfrak{M}$ -泛的，如果每个$X\in\mathfrak{M}$都可以等距嵌入$Y$中，但不存在满足这个性质的$Y$真子集。我们发现的条件下，对于给定的度量空间$X$，有一类$\mathfrak{M}$的度量空间，使$X$是最小的$\mathfrak{M}$ -泛。我们推广的概念，最小$\mathfrak{M}$ -普遍度量空间的概念，最小$\mathfrak{M}$ -普遍类的度量空间，并证明了唯一性，直到同构，为这些类。给出了属于一类$\mathfrak{M}$的度量空间的不交并是极小$\mathfrak{M}$ -泛的充要条件，并对三点度量空间和$n$维赋范空间类构造了极小泛度量空间的例子.此外，对于度量空间类$X$的某些子类，还找到了极小泛度量空间，它具有下列性质.在$X$的每三个不同点中，有一个点位于其他两个点之间。
英文摘要：
Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions under which, for given metric space $X$, there is a class $\mathfrak{M}$ of metric spaces such that $X$ is minimal $\mathfrak{M}$-universal. We generalize the notion of minimal $\mathfrak{M}$-universal metric space to notion of minimal $\mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class $\mathfrak{M}$ is minimal $\mathfrak{M}$-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces $X$ which possesses the following property. Among every three distinct points of $X$ there is one point lying between the other two points.
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