关于由Fuzzy度量导出的度量的几点注记
Some remarks on metrics induced by a fuzzy metric
摘要：
我们引入了一个清晰度量$d_M$作为两个不同网$(Δ_{M,λ})$的公共极限和由一个模糊度量$M$引起的清晰度量的公共极限$(δ_{M,λ})$，并证明了这些极限中的每一个的存在性等价于另一个极限的存在性，并且它的特征在于原始模糊度量$M$上的另一个条件.我们还得到了这些近似度量$Δ_λ$和$δ_λ$的一些性质。另一方面，对于给定的明确度量$d$，建立表示具有$\{0,1\}$和$d$中的值的$M_d$的模糊度量与相同的拓扑兼容。进一步证明了如果一个清晰度量$d$诱导一个模糊度量$M_d$，则由该模糊度量诱导的所有近似清晰度量$Δ_{M,λ}$和$δ_{M,λ}$都等于原始度量$d$ .
英文摘要：
We introduce a crisp metric $d_M$ as the common limit of two different nets $(Δ_{M,λ})$ and $(δ_{M,λ})$ of crisp metrics induced by a fuzzy metric $M$ and prove that the existence of each of these limits is equivalent to that of the other and it is characterized by another condition on the original fuzzy metric $M$. We also derive some of the properties of these approximate metrics $Δ_λ$ and $δ_λ$. On the other hand, for a given a crisp metric $d$, establish that the fuzzy metric representing $M_d$ with values in $\{0,1\}$ and $d$ are compatible with the same topology. Further, we prove that if a crisp metric $d$ induces a fuzzy metric $M_d$, then all the approximate crisp metrics $Δ_{M,λ}$ and $δ_{M,λ}$ induced by this fuzzy metric are equal to the original metric $d$.
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