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Given a set $D$ defined by

$$D = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 < x + y + z\},$$ Show that $D$ is an open subset of $\mathbb{R}^3$. I have to use the fact that the pullback image of an open set is open, and I have to use the fact that a union of open sets is an open set. I am not sure where to start.

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    $\begingroup$ For a start, consider $f(x,y,z)=x^2 + y^2 -(x + y + z)$. Is $f$ continuous? $\endgroup$ – Shivering Soldier Mar 9 at 12:37
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With $f(x,y,z)=x^2 + y^2 -(x + y + z)$ we have

$$D= f^{-1}( ( - \infty,0)).$$

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