Topologically Equivalent Metrics¶

Definition¶

Two metrics $d_p$ and $d_q$ on a space $X$ are termed (topologically) equivalent if they produce identical topology (the set of open subsets of $X$) on $X$, i.e., a subset $E \subseteq X$ is open with respect to $d_p$ if and only if it's open under $d_q$. This relationship is expressed as $d_p \sim d_q.$

Interpretation¶

Two metrics are topologically equivalent if they determine the same collection of open sets. They may measure distances differently, but they agree on which sets are open, and therefore on every notion that is defined through open sets.

In the left panel, the round $d_p$-ball $B_{d_p}(\boldsymbol{x},r)$ is $d_p$-open, and around its point $\boldsymbol{y}$ fits a diamond-shaped $d_q$-ball $B_{d_q}(\boldsymbol{y},\delta)$, so $\boldsymbol{y}$ is an interior point of that ball for $d_q$ as well. The right panel does the same with the roles swapped. If this works around every point of every ball, a set is $d_p$-open if and only if it is $d_q$-open: the two metrics give the same topology.

The radius $\delta$ may depend on the point and on $r$; no fixed ratio between the metrics is required. This is the difference from strongly equivalent metrics: there $d_q\le b\,d_p$ gives $B_{d_p}(x,r/b)\subseteq B_{d_q}(x,r)$ with the same factor everywhere, which is why strong equivalence implies topological equivalence. On $\mathbb{R}^n$ all $p$-metrics are strongly equivalent, so "open" in $\mathbb{R}^n$ does not depend on which of them is used.

Topologically equivalent metrics have the same convergent sequences. A common slip is to assume that they also agree on how large distances are. On $\mathbb{R}$, $\min(|x-y|,1)$ gives the same open sets as $|x-y|$$y|$, although under it no two points are more than $1$ apart. Similarly, $x_n=\tfrac{\pi}{2}-\tfrac{1}{n}$ is a Cauchy sequence for $|x-y|$ on $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ but not for $d_{\tan}$ from the Examples.

Key Results¶

• If two metrics are strongly equivalent, then they are topologically equivalent.

Examples¶

• On the interval $(-\pi/2,\pi/2)\text{,}$ the metric $d_{\tan}(x,y)=|\tan x-\tan y|$ is topologically equivalent to the absolute value metric, but they are not strongly equivalent: the ratio $|\tan x-\tan y|/|x-y|$ is unbounded near $\pm\pi/2\text{.}$
• On $\mathbb{R}^n$, the discrete metric is not topologically equivalent to any $p$-metric: every singleton is open for the discrete metric, but not for a $p$-metric.

Prerequisites¶

• Topology and Topological Space
• Open and Closed Sets in a Metric Space

Related Concepts¶

• Strongly Equivalent Metrics
• Equivalent Norms

Sources¶

• Math III: Analysis · Lecture 6

Aliases: topologically equivalent Symbols: delta, pi, real numbers, subset

Last updated: 27 October 2026 11:52