Closedness of $Q^3=\{(x,t)\in\mathbb{R}^2\times\mathbb{R}:\|x\|_2\leq t\}$

Question

Let the set $Q^3=\{(x,t)\in\mathbb{R}^2\times\mathbb{R}:\|x\|_2\leq t\}$.

In my math notes, it is written that the proof of the closedness of $Q^3$ "follows from continuity of the $\ell_2$ norm." Can someone explain the connection between the norm and the set being closed?

Answer 1

Since that norm is a continuous map, the map$$\begin{array}{rccc}f\colon&\mathbb{R}^2\times\mathbb R&\longrightarrow&\mathbb R\\&(x,t)&\mapsto&\lVert x\rVert_2-t\end{array}$$is continuous. And $Q^3=f^{-1}\bigl((-\infty,0]\bigr)$. So, since $(-\infty,0]$ is closed, $Q^3$ is closed too.