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:


1. Are there well known examples of sets for which $T \subseteq U$ but $N\_r(T) > N\_r(U)$?
2. 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). 



When is the internal covering number of a metric space monotonic?
The most obvious example: 
Suppose $U$ is a closed ball of radius $1$ in $\mathbb R^d$, and $T$ is the corresponding sphere. Then if $r = 1$, $N\_r(U) = 1$ but $N\_r(T) > 1$.


EDIT:
Let $X$ be any metric space such that there are three points $a,b, c$ with $d(a,b) \le d(a,c) < d(b,c)$. Then take $d(a,c) \le r < d(b,c)$, $T = \{b,c\}$ and $U = \{a,b,c\}$. We have $N\_r(U) = 1$ but $N\_r(T) = 2$. 
Asking that this example does not exist is a rather severe restriction on the metric space!


