compact $\Leftrightarrow$ sequentially compact $\Leftrightarrow$ totally bounded and complete

Question

Let $X$ be a metric space. Assuming the axiom of countable choice, the following are equivalent:

1. $X$ is compact.
2. $X$ is sequentially compact.
3. $X$ is totally bounded and complete.

What if we don't assume the axiom of countable choice?

I think $1\Rightarrow 3$ holds, but I don't know the rest.