Let $(X,d)$ be a metric space with finite Assouad dimension $0<C_X$. It seems intuitive to me that if $\emptyset \subset Y\subseteq X$ then $Y$ is also doubling and its Assouad dimension, denoted here by $C_Y$, should satisfy $C_Y\leq c C_X$ (where $c$ is some absolute constant independent of $X$ and of $Y$).
Is this true, and if so where can I find this fact?
Best Answer
This is Lemma 9.6(i) in J. C. Robinson, Dimensions, embeddings, and attractors. Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.
In the proof the author says "it is obvious". I am no longer sure if it is obvious (perhaps it is) since I see a potential issue: if $Y\subset X$, then every ball in $Y$ is a restriction of a ball from $X$, but if $B$ is a ball in $X$ not centered at $Y$, then $B\cap Y$ is not a ball in $Y$ so there are less balls in $Y$ to cover. Is it an issue? I haven't checked (I still did not have my morning coffee).