Continuous inclusion of metric spaces of smaller capacity

Question

If $(X,d_X)$ is a compact metric space, and $(Y,d)$ is another metric space. Moreover, suppose that the metric capacity of $(Y,d)$ is at-least that of $(X,d_X)$, that is $$ \kappa_X(\epsilon)\leq \kappa_Y(\epsilon) ; (\forall \epsilon \in (0,1] . $$

Here the metric capacity of a metric space $(X,d_X)$ is defined by $$ \kappa_X(\epsilon)\triangleq\sup\left\{ k : \exists x_0,\dots,x_k \in X : \exists r>0, \sqcup_{i=1}^k B(x_i;\epsilon r)\subseteq B(x_0;r) \right\} $$

Does there necessarily exist a continuous injection of $(X,d_X)$ into a metric ball in $(Y,d)$ of some (finite) radius?

Note:

• I'm not looking for Lipschitz, just continuous inclusion.
• $X,Y\neq \emptyset$.

Answer 1

The answer is no. You can have $X$ path connected and $Y$ totally disconnected satisfying your condition. Then any continuous map from $X$ to $Y$ is necessarily constant so there are no injective maps from $X$ to $Y$.

Here is another example: $X=S^1$ is a unit circle of with the geodesic metric and $Y=\mathbb{R}$. Since $X$ and $Y$ are locally isometric, it follows that $$ \kappa_X(\epsilon)\leq \kappa_Y(\epsilon) ; (\forall \epsilon \in (0,1]). $$ Indeed, if $r\le\pi$ in the definition of $\kappa$ the corresponding quantities are equal (balls of radius $r$ are isometric balls). If $r>\pi$, then the quantity in the definition of $\kappa_X(\epsilon)$ is less than the corresponding one in the definition of $\kappa_Y(\epsilon)$ (ball in $Y$ is bigger - segment of length $2r$ than the ball in $S^1$ - the unit circle and they both are locally isometric).

However there is no injective map from $X=S^1$ to $Y=\mathbb{R}$.