TITLE: Why do I think Lebesgue’s number lemma is wrong...
QUESTION [5 upvotes]: Counterexample: $[0,1]\times[0,1]$ with induced subspace topology from $\mathbb{R}^2$ is compact. The open cover $\mathscr{U}$ is just the two circular sectors. When we look at the up-left corner and down-right corner, it fails - there is no such $\delta$, such that let the open ball be only in one of  circular sectors... what is wrong?

REPLY [8 votes]: The upper-left and bottom-right corners are not contained in either open circle. That is, your claimed counterexample fails to cover the square.
Moreover, I think you may have misunderstood what is important about Lebesgue's number lemma. The following statement

Let $\mathcal U$ be a cover. For any $x$, there is some $\delta$ and some $U\in \mathcal U$ such that $B(x,\delta)\subseteq U$ 

is clearly true of any cover - no need for compactness. This is basically just writing down what "open" and "cover" are defined to be. Given that your counterexample fails this condition, it is clear that it is not an open cover.
Note that Lebesgue's number lemma exchanges the quantifiers and says that there is some $\delta$ that works for all $x$. This is what requires compactness.