论文标题
扩展适当的指标
Extending proper metrics
论文作者
论文摘要
我们首先证明了Tietze-urysohn定理的版本，用于在$σ$ -Compact本地紧凑的Hausdorff Space上定义的非负实数的正确功能。作为其应用程序，我们证明了适当指标的扩展定理，该定理指出，如果$ x $是$σ$ - compact局部紧凑的空间，$ a $是$ x $的封闭子集，而$ d $是$ a $ a $ a $ a $ a $ a $ a $ a $的$ a $的适当度量，那么$ a $的$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ $ d | _ {a^{2}} = D $。此外，如果$ a $是适当的撤回，我们可以选择$ d $，以便$（a，d）$是准时为$（x，d）$的准代表。我们还显示了上面关于超级空间的定理的类似物。
We first prove a version of Tietze-Urysohn's theorem for proper functions taking values in non-negative real numbers defined on $σ$-compact locally compact Hausdorff spaces. As its application, we prove an extension theorem of proper metrics, which states that if $X$ is a $σ$-compact locally compact space, $A$ is a closed subset of $X$, and $d$ is a proper metric on $A$ that generates the same topology of $A$, then there exists a proper metric on $X$ such that $D$ generates the same topology of $X$ and $D|_{A^{2}}=d$. Moreover, if $A$ is a proper retraction, we can choose $D$ so that $(A, d)$ is quasi-isometric to $(X, D)$. We also show analogues of theorems explained above for ultrametric spaces.
