Explanations of Lebesgue number lemma

Solution 1:

The intuition is the following : for each $x$ of $X$ is included in at least one $U$ of $\mathcal U$. Since $U$ is open, it contains a ball centered at $x$ and with positive radius, right ? Let $r(x, U)$ denote the supremum of all such radius. And let $r(x)$ the supremum of all the $r(x,U)$, with $U\in\mathcal U$.

So $r(x)$ is a continuous positive function on $X$. But if $X$ is not compact, you may find a sequence $(x_n)$ such that $r(x_n)$ tends to zero.

However, if your metric space $X$ is compact, then $r$ shall have a minimum. This minimum is of course positive since $r$ is. This minimum is a Lebesgue number, the greater one.

And in fact, the intuition is a proof !

A covering with positive Lebesgue number is called a uniform covering.

Solution 2:

The characteristic property of a compact space is that every open cover has a finite subcover, which is used in proving the Lebesgue number lemma. I'm sure you've read a proof of the Lebesgue number lemma somewhere; otherwise I would recommend reading this one, which I found to be quite intuitive. Quickly, the idea of the proof is as follows: Start off with a given cover $\mathcal{U}$. For every point in $X$, there is an $\epsilon$ neighborhood contained in an element of $\mathcal{U}$. These neighborhoods form an open cover, which must have a finite subcover since $X$ is compact. Now if we pick $\delta$ to be $1/2$ times the least $\epsilon$, we can show that it satisfies the requirements of being a Lebesgue number.

I also rather like fer the proof by contradiction: suppose there was no such $\delta$. In that case, the negation of the Lebesbue number lemma says: there exists a cover $\mathcal{U}$ such that for every $\delta>0$, there exists an open ball of radius $\delta$ not contained in any element of $\mathcal{U}$. So, for each $n$ , we can choose an $x\in X$ such that no $U$ contains $B_{1/n}(x_n)$ . Now, $X$ is compact so there exists a subsequence $(x_{n_k})$ of the sequence of points $(x_n)$ that converges to some $y\in X$. But since $\mathcal{U}$ is an open cover, there is some $\gamma$ such that $B_\gamma(y) \subseteq U$ for some $U\in \mathcal{U}$. Pick $k$ large enough so that $|x_{n_k}-y|\leq \gamma/2$ and $1/n_k < \gamma/2$. Use the triangle inequality to show that $B_{1/n_k}\subseteq U$.

In general, my (not entirely accurate) intuition for this is that for compact spaces, finite covers are sufficient. If a Lebesgue number did not exist, we would get a sequence of sets which are not included in the elements of the cover, which would contradict the finiteness of the cover.

As for your followup questions:

1. Any number smaller than a Lebesgue number is also a Lebesgue number.
2. As you saw above, compactness shows up prominently in the proof for the lemma.

Solution 3:

A non-compact counterexample: consider the open cover of $\mathbb{R}$ $$ \{(n,n+1) : n\in\mathbb{Z}\}\cup\{(n-\epsilon_n,n+\epsilon_n) : n\in\mathbb{Z}\} $$ where $\epsilon_n\to0$ as $|n|\to\infty$. Balls centered at integer points need to get smaller and smaller the further you are from the origin, so a Lebesgue number doesn't exist.

Also note that a number smaller than a Lebesgue number is a Lebesgue number.

Solution 4:

For 2.:

Let $\cal U$ be an open cover of the compact space $X$. Suppose that for each positive integer $n$, the open ball $B_{1/n}(x_n)$ is not contained in any element of $\cal U$. Since $X$ is compact, it is sequentially compact, and hence the sequence $\{x_n\}$ has a limit point $x$. Let $U$ be an element of $\cal U$ containing $x$. Since $U$ is an open set, there is an open ball $O\subset U$ with $x\in O$. For sufficiently large $n$, we have $$ B_{1/n}(x_n)\subseteq O\subseteq U, $$ which contradicts the selection of the $x_n$.

I'm not sure how to address 1.); but this may help to understand what motivates the Lebesgue number:

One way to prove that a sequentially compact space is compact (note in the above, one need only assumed $X$ is sequentially compact) is to use the concept of Lebesgue number and total boundedness of $X$. Both of these properties are enjoyed by sequentially compact metric spaces.

Given an open cover of the sequentially compact space $X$, one can cover it with a finite number of balls of radius less than half of the Lebesgue number. These balls are contained in elements of the open cover; so the open cover has a finite subcover.

One way to think about the Lebesgue number is that if $\cal U$ has Lebesgue number $\delta$, then there are elements of $\cal U$ that cover "big chunks" of $X$ (namely open balls with radius less than $\delta$). Contrast this with the open cover of $(0,1)$ $$ {\cal U} =\{ (\textstyle{1\over n+1},{1\over n-1} ) :n=2,3,\ldots \} $$ This open cover is "small" in the sense that the open ball $(0,\epsilon)$ is contained in no element of it.

Solution 5:

1. Here is yet another explanation of the Lebesgue Number Lemma: It says that for compact metric spaces we have a very nice extension of the notion of an open set to an open cover. Indeed, Lebesgue Number Lemma guarantees that for a compact metric space $X$ we have that $\exists \delta>0, \forall x\in X, \exists U_x\in \mathcal{U}: B_\delta(x)\subseteq U_x$. This means that the openness condition of open sets holds for the cover (this is trivial), and it holds "set-by-set". So it does not happen that we can find an open ball separated by the sets in the cover (and there is no other set in the cover that contains it).

As a side note, we can also use this lemma to define an equivalence relation on the open balls of $X$, once we make the cover disjoint. Though I am guessing this might turn out to be a chore to pursue.

2. As pointed in the above answers, compactness prevents taking limits so that $\delta$ turns out to be $0$, in which case the balls are only points, and the result is trivial.

3. Yes.