Let $X$ be a vector space, equipped with two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ Which are equivalent.
What is the easiest way to prove that these two equivalent norms induce same topology?
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1By showing that any open ball under norm 1 contains an open ball under norm 2 and vice versa. – Steve B Jun 25 '18 at 10:47
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I don't mean to oversimplify, but the definition I just found of equivalent norms is that they induce the same open sets, in which case this fact is immediate... – Jun 25 '18 at 12:59