I was calculating a domain of a function $f(x,y)$ and I need to say if the domain is an open set or closed set, and if it is bounded.
At the end of my calculations, I got $xy \geq 1$, which is the correct domain.
The final answer in the book said it is closed and not bounded.
I wanted to ask you guys, how can a set of point be infinite and still be closed ?
From single variable calculus, I know that for example $[a, +\infty)$ is an open set, since we infinity can't be equal to anything, so why is it different with two variables. And if there is a mistake and it is not closed, than what's the difference between bounded and closed then ?
Thank you !
Best Answer
From the axioms of topology, a set is closed iff its complement is open, and in your case the complement $\{(x,y)\in\mathbb{R}^2\mid xy < 1\}$ is open (eg, you can see it as the reciprocal image of the open set $(\infty, 1)$ by the continuous function $(x,y)\mapsto xy$).
Note that boundedness and being open/close are completely independent: $\mathbb{R}$ is by definition an open in $\mathbb{R}$ (endowed with its usual topology) (and also a closed set, btw), so is $\emptyset$. $(0,1)$ is bounded but open, $[0,1]$ is bounded but closed, $(0,1]$ is bounded but neither open nor closed.
Edit: incidentally, $[a,\infty)$ is a closed set: its complement is $(-\infty,a)$ which is open. If you need, I can give you pointer to several (equivalent) definitions of open/close, which may help you in your understanding.