A Totally Bounded Metric Space is Separable

Let $(X,d)$ be a metric space and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$, a topology on $X$ by Metric Open Sets Form a Topology.

Then the following hold.

1. (Separability.) If $X$ is totally bounded in $(X,d)$, then there is a countable subset $D\subseteq X$ that is dense in $X$ for $\mathcal{T}_d$; that is, $(X,d)$ is separable.

2. (Compact and sequentially compact spaces.) If $X$ is compact in $(X,\mathcal{T}_d)$, or sequentially compact in $(X,d)$, then $X$ is totally bounded in $(X,d)$, and consequently $(X,d)$ is separable.