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I am reading the proof in Peter Lax book (page 169). By baire category we have some set $MB_i$ dense in some open set U. Then they translate those to origo by the form $M(B_n -x_0)$. And state that $B_n - x_0$ is inside the ball of radius $n + |x_0|$, this I can buy. But then he conclude by homogenity that $MB_1(0)$ is dense in $B_r(0)$. How is this certain? If we look at $MB_{n+|x_0|}(0)$ how do we know that we are not bigger then the translated U?

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Deciphering "by homogeneity": for any $\lambda>0$ the map $x\mapsto \lambda x$ is a homeomorphism of the space. Therefore, $A$ is dense in $B$ if and only if $\lambda A$ is dense in $\lambda B$. // Also, being bigger is not a problem; I interpret "$A$ is dense in $B$" as $\overline{A}\supset B$, rather than $\overline{A}=B$. – user53153 Jan 15 '13 at 22:08

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