Using the closed set formulation of continuity (Theorem 18.1), show that the following are closed subsets of \(\mathbb{R}^2\):
\[ A = \ts{x \times y \mid xy = 1}, \]
\[ S^1 = \ts{x \times y \mid x^2 + y^2 = 1}, \]
\[ B^2 = \ts{x \times y \mid x^2 + y^2 \leq 1}. \]
The set \(B^2\) is called the (closed) unit ball in \(\mathbb{R}^2\).
Solution
The coordinate projections \(p_1,p_2:\mathbb R^2\to\mathbb R\) are continuous. Hence so are \[ F(x,y)=xy=p_1p_2, \qquad G(x,y)=x^2+y^2=p_1^2+p_2^2. \] Since singletons are closed in \(\mathbb R\), \[ A=F^{-1}(\{1\}) \] is closed. Likewise \[ S^1=G^{-1}(\{1\}) \] is closed. Finally \((-\infty,1]\) is closed in \(\mathbb R\), so \[ B^2=G^{-1}(( -\infty,1]) \] is closed. This proves all three claims directly from the closed-set formulation of continuity.