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How do you show the complement of these subsets are open?

  1. $S^1=\{(x_1,x_2)\in\Bbb R^2\mid x^2_1+x^2_2=1\}$
  2. $B^c_1=\{(x_1,x_2)\in\Bbb R^2\mid x^2_1+x^2_2\ge1\}$
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    Try to prove the corresponding versions of the statements for $\mathbb{R}^1$ first. How do you prove that $S^1 = \{ x \in \mathbb{R} : x^2 = 1 \}$ and $\{ x \in \mathbb{R} : x^2 \geq 1 \}$ are closed (i.e. their complements are open)? You pick a point $x$ in the complement and find a sufficiently small ball of radius $\epsilon > 0$ centered at $x$ so that the entire ball is contained in the complement... Hope that helps get you started.2012-03-01
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    @kahen: Edited. Please go easy on folks who are new to this site.2012-03-01
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    Looks like $x21$ is a typo. I suppose you mean $x12$.2012-03-01
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    @Patrick: I guess he copy-pasted from a HTML or PDF exercise sheet, accidentally missing the superscript/subscript formatting; the original must have been produced by software that puts the superscript first on general principles.2012-03-01

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