Let d1 and d2 be metrics for a set X, and let B1 and B2 denote, respectively, the families of all open balls generated by d1 and d2. Show that d1 and d2 are equivalent metrics iff B1 and B2 are equivalent bases.
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1Usually, posts with just the question are not taken well in this website. Try to write what you understand of the problem, which definitions confuse you, and attempts! After this, I'm sure we will help you! – grover Mar 30 '22 at 19:26
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Start by writing out the definition of "equivalent metrics" and the definition of "equivalent bases" and comparing the two. – Paul Sinclair Mar 31 '22 at 14:24