[Def]: Equivalent Metric

Let $d_1$ and $d_2$ be two metrics on a nonempty set $X$.

We say that $d_1$ and $d_2$ are equivalent $d_1\sim d_2$ if for any $x_0\in X$ and $\varepsilon>0$

1. There exists $\delta_1>0$ s.t. if $d_1(x_0,y)<\delta_1$, then $d_2(x_0,y)<\varepsilon$
2. There exists $\delta_2>0$ s.t. if $d_2(x_0,z)<\delta_2$, then $d_1(x_0,z)<\varepsilon$

給定 $\varepsilon$, 一定可以找到 $\delta_1$, 在 $d_1$ 視角下距離小於 $\delta_1$ 的這些點, 在 $d_2$ 視角下都會小於給定的 $\varepsilon$

反之也是

要理解的話搭配 Equivalent Metric Lemma