= Metric topology
The <metric topology> consists of the <sets> $U$ such that, for every $x\in U$, some <open ball> $B_d(x,r)$ lies in $U$. Thus <open sets> describe which sufficiently small changes remain inside a region; they do not specify the numerical size of those changes.
For $v\in\mathbb R^2$, the <L1 norm>, <Euclidean norm>, and <maximum norm> satisfy
$$
\|v\|_\infty\leq\|v\|_2\leq\|v\|_1\leq\sqrt2\,\|v\|_2\leq2\|v\|_\infty.
$$
These bounds make them <equivalent norms>. For example, $B_\infty(a,r/2)\subseteq B_1(a,r)\subseteq B_\infty(a,r)$: every <open ball> contains a sufficiently small <open ball> for either other <metric>. Hence all three define the same <open sets>, <convergent sequences>, and <continuous maps>. More generally, any two <norms> on a real or complex <finite-dimensional vector space> are <equivalent norms>; see https://www.maths.ox.ac.uk/system/files/attachments/B4.1%20Lecture%20Notes.pdf[H. A. Priestley, Functional Analysis I, §§0.5–0.7 and Theorem 4.3].
The <metric topology> therefore does not determine numerical distances. Multiplying a <metric> by $100$ preserves its <open sets> while multiplying every distance by $100$. Nor does the general <equivalence of metrics> require constant-factor bounds: that stronger comparison holds for the three <norms> used here.
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