I read somewhere that $xy=1$ is a closed set in $\Bbb{R}^2$.
A closed set is defined as the complement of an open set, or one which contains all its limit points. In metric spaces, it is defined as the complement of the union of balls $B(x,\epsilon)$, where $\epsilon>0$. For example, $(-\infty,0)$ is open in $\Bbb{R}$ as it is $\bigcup_{i=-1}^{-\infty}B(i,1)$, where $i\in \Bbb{Z}^-$.
Is $xy=1$, or any graph for that matter, closed because we can take any point $( x,y)$ outside of it and draw an open set centred on it such that $B((x,y),\epsilon)$ lies completely inside the complement of the graph, thus saying the complement is the union of open sets (hence open)?
Thanks in advance!
Best Answer
More simply, note that $(x,y)\mapsto xy$ is a continuous function $\Bbb R^2\to\Bbb R,$ and that $\bigl\{(x,y)\in\Bbb R^2:xy=1\bigr\}$ is the preimage of the closed set $\{1\},$ so is closed by continuity.