Show that on any uncountable set $X$, there is a metric on $X$ with respect to which $X$ is not separable

Question

Show that on any uncountable set $X$, there is a metric on $X$ with respect to which $X$ is not separable

The definition we are using is $X$ is separable if $\exists E$ dense in $X$ and set $E$ is countable

Answer 1

Take the discrete metric. Then the only dense subset of $X$ is $X$ itself, which is not countable.