For an open covering of a compact metric space $(X, d_X)$ there exists a real number $\delta > 0$ such that for any subset with diameter less than $\delta$, there is an element of the covering that contains it. This number is called the Lebesgue number for the covering.

Obviously, any number smaller than the Lebesgue number is still a Lebesgue number for that covering, so there is an interval of the real numbers: $$(0,\delta_m]$$ This interval contains all Lebesgue numbers for a specific covering.

Consider $A\subset \Bbb{R}^2$ and the covering by open balls: $$\mathcal{B} = \{ B_{d}(x, 1/2), x\in A\} $$ The $\delta_m$ in this case is exactly $1/2$. Any set bigger than that in diameter would not fit inside any element of $\mathcal{B}$. However, this proves a little more difficult when we select a finite subcover, and $\delta_m$ depends heavily on the way we choose it.

My question is: given an open covering, is it always possible to find $\delta_m$ explicitly?

My guess is, given a finite open covering $\{A_i\}_{i=1...n}$ of a subset of a compact metric space $(X, d)$ and given the function: $$f(x) = \frac{1}{n}\sum_{i=1}^{n} d(x, X-A_i)$$ Then we have: $$\delta_m = \min[f(x)]$$ But I'm unsure this is the right approach. Its impracticality and unsuitability to be generalized to infinite open coverings suggest me there could be a much more elegant way to find the number.


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