Which two would be an example of non-equivalent metrics on $\mathbb{R}$.

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Additionally, it's required that neither of them induces the discrete topology.

what all metrics do you know on $\mathbb{R}$? – Jan 24 '14 at 09:59 — , Jan 24 '14 at 09:59

score 8 · Accepted Answer · answered Jan 24 '14 at 10:06

8

The following metric is not topologically equivalent to the standard Euclidean metric:

$$d ( x , y ) = \begin{cases} \min \{ | x - y |,1\}, &\text{if }x,y\text{ are either both rational or both irrational} \\ 1, &\text{otherwise} \end{cases}$$

(The underlying idea can be used to generate lots of inequivalent metrics on $\mathbb{R}$.)

answered Jan 24 '14 at 10:06

user642796

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Thank you, that was an interesting example. – Matias Heikkilä Jan 24 '14 at 10:27

Which two would be an example of non-equivalent metrics on $\mathbb{R}$.

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