Does there exist a metric on $X$ such that $X^{∗} − X$ is uncountable?

Question

Let $X^{∗}$ denote the completion of the metric space $X$. Is there a metric on the open interval $X = (0, 1)$ generating the Euclidean topology for which $X^{∗} − X$ is uncountable? Any idea and hint would be helpful. Thanks!

Answer 1

Consider the "topologist's sine curve" in the form $$C=\{(x,\sin(1/x):0<x<1\}.$$ This is a subspace of $\Bbb R^2$ homeomorphic to $(0,1)$, so we can think of $(0,1)$ as a metric space by considering the metric of $C$ inside $\Bbb R^2$. Thus $$d(x,y)=\sqrt{(x-y)^2+\left(\sin\frac1x-\sin\frac1y\right)^2}.$$ The closure of $C$ in $\Bbb R^2$ is the completion of $C$ with respect to this metric, and contains the uncountable set $\{(0,y):-1<y<1\}$. Thus $C^*-C$ is uncountable.