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Let X be a metric space with metric d. I'm trying to show that U(xo;e) is an open set. What I note so far is to talk about a subset U that for each xo an element of U, there is a corresponding e > 0 s.t. U(xo;e) is contained in U.

Does this finish the proof?

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    What does this have to do with "R^n analysis"? And what is your exact definition of U(x0;e)?2011-11-06
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    What you write does not make sense. You are trying to prove a statement about x0, but your argument uses x0 as a free variable (when you say "for each x0 an element of U").2011-11-06

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