I had a question of curiosity. Take the interval $(0,1)$ with the usual metric in $\mathbb{R}$. Is it possible to find closed sets $X$ and $Y$ with $X\cap Y=\varnothing$ such that there is a sequence $\{x_n\}$ in $X$ and $\{y_n\}$ in $Y$ where $\lim_{n\to\infty}\vert x_n-y_n\vert=0$?
I'm having a hard time picturing this, for I see closed sets as closed intervals (or finite unions of them) on the real line. For the limit to approach $0$, I intuitively see that the points eventually all bunch up near some point. With open intervals, I think you could take intervals like $X=(a,b)$ and $Y=(b,c)$ with $0
Is there a way to get around this snag?