Total Boundedness of Set in Metric Subspace

Question

Let $(X, \rho)$ be a metric space and let $(Y, \sigma)$ be subspace of $X$ where $\sigma$ is the metric inherited from $\rho$. Now, suppose we have a set $C \subset Y$ which is totally bounded in the topology of $Y$, i.e. given an $\varepsilon > 0$, we can find points $x_{1}, \dots, x_{n}$ in $Y$ such that $$ C \subset \bigcup_{i=1}^{n} B^{\sigma}_{\varepsilon}(x_{i}) $$ where $ B^{\sigma}_{\varepsilon}(x_{i})$ is the open ball of radius $\varepsilon$ with respect to the metric $\sigma$. Given this, how do I prove that $C$ it toally bounded in the topology of the parent space $X$? For each of the $x_{i}$ above, I could pick an open set $O_{i}$ in $X$ and write $B_{\varepsilon}^{\sigma}(x_{i}) = O_{i} \cap Y$, but I'm not sure where to proceed with this (i.e. how do I get finitely many balls open in $X$?)

Answer 1

The same points that work in $Y$ work in $X$, as $Y \subseteq X$ and $B^\sigma_\epsilon(y) = B^\rho_\epsilon(y)\cap Y$ for $y \in Y$ and $\epsilon>0$. There are no choices necessary.