Given a radius $r > 0$, the internal covering number of a subset $T$ of a metric space $(X, d)$ is denoted $N_r(T)$ and is defined to be the smallest number of balls of radius $r$ (under $d$) with centres in $T$ such that $T$ is contained in the union of the balls.

Given another subset of $X$, $U$, which is a superset of $T$, it is not necessarily true that $N_r(T) \leq N_r(U)$.

My question:

- Are there well known examples of sets for which $T \subseteq U$ but $N_r(T) > N_r(U)$?
- Are there necessary/sufficient conditions on $X$ or $d$ such that the internal covering number is monotonic, i.e. $T \subseteq U \implies N_r(T) \leq N_r(U)$?

In case it is relevant, my application is to cases in which $X$ is generated by $T$ under some (infinite) set of transformations (e.g. a Lie group).

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