The set is given as subset $X = \{(x, y) | x > 0, y > 0\}$ of $\mathbb{R}^2$. The set is open. Find an open cover for the set that does not admit a finite subcover. The purpose is to show that the set is compact directly without referencing the Heine-Borel property.
If I find the open cover with no finite subcover, then it shows that the set is not compact by definition. I understand the idea of the question, but I'm not quite sure how to go about actually finding an open cover.
Is there a beginner process I could follow? I've basically just been guessing without any luck. What intuition goes behind this and how do I check that I have the correct open cover?
Thank you!
Best Answer
"Beginners" get given problems like this often enough. The goal is to drill the idea of open covers and compactness.
But there is a secret: they are all the same! The other secret is that "beginners" do not see that they are all the same until they have drilled repetively enough so that it sinks in. It is tempting to fuss about the set given to you and worry it to death. But you can use the exact same argument for every set given.
Idea 1. In $\mathbb R^2$ (or any metric space) construct the expanding sequence of open balls centered at $(0,0)$ and with radius $n=1,2,3, \dots$ $$O_n= \{ (x,y): x^2 + y^2 < n\}.$$
This collection $\{O_n\}$ is an open cover of every set. There is no finite subcover of any unbounded set.
Idea 2. In $\mathbb R^2$ (or any metric space) construct the contracting sequence of closed balls centered at $(0,0)$ and with radius $\frac1n$, for $n=1,2,3, \dots$ $$C_n= \left\{ (x,y): x^2 + y^2 \leq \frac1n\right\}.$$
Take complements: $U_n =\mathbb R^2 \setminus C_n$.
This collection $\{U_n\}$ is an expanding sequence of open sets that covers every set that does not contain the point $(0,0)$. There is no finite cover of any set that has $(0,0)$ as a limit point.
Standard baby analysis problem: Show that this set $ E = \dots$ in $\mathbb R^n$ is not compact by constructing a cover of $E$ by a family of open sets that contains no finite subcover of $E$.
Hint 1: If $E$ is unbounded copy Idea #1. Pay no attention to $E$ other than to note that it is unbounded.
Hint 2: If $E$, unfortunately, is bounded, then find a limit point of $E$ that is not a member of $E$. Pay no attention to $E$ other than to note that it does not contain that limit point. Copy Idea #2 using that limit point in place of $(0,0)$ in the idea.
This works in $\mathbb R^n$ because a set there is compact if and only if it is bounded and contains all of its limit points (i.e., is closed). In a general metric space it may be the case that some closed, bounded sets fail to be compact.