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This question already has an answer here:

how we can give an example of two metrics on a space that induce the same topology , but are not equivalent!(so do we can generalize our definition about equivalent of metrics?)

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marked as duplicate by Lee Mosher general-topology Jun 19 at 15:34

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Let $(X,d)$ be a metric space with $d$ unbounded and consider the metric $d'(x,y)=\min\{d(x,y),1\}$. This induces the same topology, but is not equivalent to $d$, since it is bounded

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  • $\begingroup$ short and sweet! $\endgroup$ – Prototank Jun 19 at 15:33

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