Equivalent Metrics

Real Analysis · Axiom Academy

Show the Euclidean, taxicab, and sup metrics on are strongly equivalent, so they share the same topology. Two metrics d_1 and d_2 on a set are topologically equivalent when they induce the same topology — the same open sets, equivalently the same convergent sequences and the same continuous functions. A sufficient (stronger) condition is strong (Lipschitz) equivalence : there exist constants with for all x,y . On , take the three metrics , (Euclidean), and (taxicab). Prove they are strongly equivalent , then read off that they have the same open sets, convergent sequences, and continuous functions. The unit ball of each metric. Because , a smaller distance gives a larger ball: the taxicab diamond the Euclidean circle the sup square. Because (here n=2 ), shrinking the square by a factor of n fits a square inside the diamond — so each ball contains a ball of the other metric. Same neighborhoods, same topology; only the shapes differ. Nice work! You showed the Euclidean, taxicab, and sup metrics on are strongly equivalent, so they all describe the same topology. Remember: Strong (Lipschitz) equivalence: two metrics are strongly equivalent if there are constants with for all x,y . The explicit chain: on , (and ), so each metric is squeezed between constant multiples of the others. Constants depend on the dimension: here C = n ; the bound is sharp — equality holds when every coordinate has the same magnitude.

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