Is every complete metric space not countable?

Question

I want to know if every complete metric space is not countable? I can't find a counterexample, so is it correct?

Answer 1

Consider a countable space with the discrete metric.

Answer 2

This is true if you add the hypothesis that there are no isolated points.

Answer 3

Take any complete metric space $X$ and any convergent sequence $(x_n)_{n\in\Bbb N}$ of element of $X$ and then the subspace$$\left\{x_n\mid n\in\Bbb N\right\}\cup\left\{\lim_{n\to\infty}x_n\right\}$$is complete and countable.

Answer 4

If a set is finite, then it being countable and it being complete are both rather trivial. One can add a countable sequence going to a limit as long as one also adds the limit. In fact, one can do so countably many times.