Metric spaces and distance functions.

Question

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

1. X has a countable dense subset

2. X is uncountably infinite and has only one limit point

3. 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)

Answer 1

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 $y\leq 1$.

Answer 2

For $(1)$, consider $(X,d)=(\mathbb{R},|.|)$, since $\mathbb{R}$ has $\mathbb{Q}$ the set of rational numbers as a countable subset.