A compact note on equivalent metrics.

Question

I am a graduate student of mathematics.While studying metric spaces,we often encounter equivalent metrics.I know the definition (that equivalent metrics induce the same topology)and also know that $d_1,d_2,d_\infty$ are equivalent on $\mathbb R^n$.But still,I am not quite used to equivalent metrics and feel shaky while doing problems.Can someone suggest some text,note or link that discusses properties of equivalent metrics(such as properties preserved by equivalent metric) along with problems on it?

Answer 1

Note that for equivalent metrics, the generated topology is the same, i. e., from a topologist's perspective, there is no difference between spaces with equivalent metrics. E. g.

• all continuous functions remain continous,
• the product topology remains the same
• in two such Hausdorff local compact topological groups the (existing) Haar measure is the same etc.