Metric spaces and distance functions. I need to provide an example of a space of points X and a distance function d, such that the following properties hold:


*

*X has a countable dense subset

*X is uncountably infinite and has only one limit point

*X is uncountably infinite and every point of X is isolated
I'm really bad with finding examples... Any help will be greatly appreciated! Thank you.
(I got the third one)

For the first question, hint: $\mathbb{Q}$ is a countable set.
For the third question, hint: think about the discrete metric on a space.
For the second question: Let $X=\{x\in\mathbb{R}\:|\: x>1 \mbox{ or }x=\frac{1}{n}, n\in\mathbb{N}_{\geq 1}\}\cup\{0\}$. Let


*

*$d(x,y)=1$ if $x> 1$ or $y>1$,

*$d(x,y)=|x-y|$ if $x\leq 1$ and $y\leq 1$.
