Is $S$ compact?

Solution 1:

Your solution is incorrect, because $S$ is not that rectangle. For example, $(1,2.938.472.985.345)$ is also an element of $S$, but it is not a part of the rectangle.

This number should also give you an idea about why exactly $S$ is not compact.

Solution 2:

S is closed, since $f(x,y)=(x-1)(x-2)(y-3)(y+4)$ is a continuous function, and you can verify that whenever you have a set of zeros $f(x,y)=0$ where $f$ is continuous, then it has to be closed (hint: take a convergent sequence of points in the set and show the limit also satisfies the equation because of the continuity). However, your set is not bounded! Every point $(1,y)$ with $y\in\mathbb{R}$ is in $S$, so your set contains a line(more than one actually). Your intuition is not bad in the sense that it does contain the rectangle, but it is much more than that, it is the union of the 4 lines given by the 4 factors, try to picture what it looks like.