A closed set generated by a formula
Task number: 3170
The set \(M \subseteq \mathbb R^3\) is defined as \[ M = \{(x,y,z) \in \mathbb R^3\colon \space x^2 + y^2 + z^2 = 17, \space xyz -xz + x + y = 3 \}. \] Show that \(M\) is a closed subset of \(\mathbb R^3\).
Resolution
Define functions \(F, G \colon \space \mathbb R^3 \to \mathbb R\) as \(F(x,y,z) = x^2 + y^2 + z^2\), \(G(x,y,z) = xyz - xz + x + y\). Then \(M = F^{-1}(\{17\}) \cap G^{-1}(\{3\})\). \(F\) a \(G\) are continuous and the sets \(\{17\}\) and \(\{3\}\) are closed. Using one of the equivalent definitions of continuity (the image of a closed set is a closed set), we get the that \(M\) is the intersection of two closed sets, and therefore closed.
