Open / Closed domain 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 !
 A: 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.
A: To figure out whether a domain is closed, ask yourself if it includes the boundary lines. If it does include boundary lines, the domain is closed. Because you found the domain to be xy≥1, you are including the boundary line xy=1. This domain is closed.
To figure out whether the domain is bounded or unbounded, ask yourself if you could draw a circle on the graph of the domain that would contain all of it. If you can't, the domain is unbounded. In this question, the domain is unbounded because it continues forever in the first and thirds quadrants. No circle you could draw would be able to "bound" the entire domain.
