Suppose that $Y\subset X$ is infinite (for example $ \mathbb{N} $, $\mathbb{R}$ or $[0, 1]$). $\left\{U_i: i\in \mathbb{N} \right\} $ is its $\delta$-cover if $\forall i\in \mathbb{N}$: $\mathrm{diam} \ U_i\le \delta $ and $Y \subset \bigcup_{i=1}^{\infty}U_i$.
Then let's denote $$ U(\delta) = \left\{ \left\{U_i : i\in \mathbb{N},\mathrm{diam}\ U_i \le \delta, Y \subset \bigcup_{i=1}^{\infty}U_i \right\} \right\}, $$ i.e. the set of all $\delta$-covers of $Y$.
How could I show (rigorously) that as $\delta \rightarrow 0$, the number of non-empty sets in each element of $U(\delta )$ goes to infinity?
I will assume that the set $Y$ is a subset of $\mathbb{R}$. Let $Y_{1}$ be an infinite countable subset of Y. We enumerate its elements by the sequence $(x_{n})_{n \geq 1}$.
We define the sequence $(\delta_{n})_{n \geq 1}$ as followed.
$\delta_{1} = |x_{1} - x_{2}|$,
$\delta_{2} = \min\{\delta_{1}, |x_{1}- x_{3}|, |x_{3} - x_{2}|\}$
and $\delta_{n} = \min\{\delta_{n - 1}, |x_{n+1} - x_{1}|, |x_{n + 1} - x_{2}|,...,|x_{n + 1} - x_{n}|\}$.
Now, show that any $\delta-$cover of Y must contain at least $n + 1$ non-empty sets when $\delta < \delta_{n}$.