What is the difference between isometrically isomorphism and homeomorphism?is an isometric mapping is continuous?
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An isometry is a map $f:X\to Y$ between metric spaces that preserves distances: $d_Y(fx, fy) = d_X(x, y)$. Such maps are automatically continuous (just use the $\delta$-$\epsilon$ definition of continuity) and injective, but they may not be surjective; an isometric isomorphism is one that's both a bijection and an isometry. The inverse $f^{-1}$ is then also an isometry and thus continuous, so $f$ is also a homeomorphism. Homeomorphisms aren't necessarily isometries; take $x\to x^3$ on $\mathbb{R}$ with the usual metric.
anomaly
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Thanks anomaly.plese explain epsilon delta . – user18719 Jul 26 '14 at 09:46
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It's the standard definition of continiuity for a metric space. See wikipedia, for example. – anomaly Jul 26 '14 at 09:54
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If we take epsilon=delta then the condition hold. – user18719 Jul 26 '14 at 10:11