# Show that $D$ is an open subset of $\mathbb{R}^3$

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.

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

With $$f(x,y,z)=x^2 + y^2 -(x + y + z)$$ we have
$$D= f^{-1}( ( - \infty,0)).$$