I want to show that the set [tex]S=\{(x,y)\in \mathbb{R}^2 | xy\neq1\}[/tex] is open.(adsbygoogle = window.adsbygoogle || []).push({});

I'm having trouble forming the open ball contained in S centered at each point in S. The idea I have is:

Let [tex]q\in S[/tex]. Then select an open ball [tex]B_r(q), r\in\mathbb{R}[/tex]. Let [tex]P=\{(x,y)\in B_r(q)|xy=1\}[/tex]. If P is empty, we are done. If P is not empty, then create an open ball [tex]B_m(q),[/tex] where [tex] m=\min\{d(q-c)|c\in B_r(q)\}[/tex] (d is the distance function). Then we are done.

However, I feel like this isn't sufficient because it has not been shown whether [tex]\{d(q-c)|c\in B_r(q)\}[/tex] actually does have a minimum, or if it contains elements tending to 0. Am I just missing a property of the reals somewhere? I hope so. Any help would be appreciated.

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# Show the openness of a set

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