the set $U =\{(x,-y) \in \mathbb{R}^2 : x=0,1,-1\text{ and }y \in \mathbb{N}\}$ is which of the following statement is True?

Question

In $\mathbb{R}^2$ with usual topology, the set $U =\{(x,-y) \in \mathbb{R}^2 : x=0,1,-1 \text{ and } y \in \mathbb{N}\}$ is which of the following statement is True ?

a) neither closed nor bounded

b)closed but not bounded

c)bounded but not closed

d)closed and bounded

I thinks it will be bounded but not closed because $(x,-y)$ is an open set that is option c) will be true

Is its True ?

any hints/solution will apprecaited

thanks u

Answer 1

Surely it is not bounded since $(0,-1)$ , $(0,-2)$ , $(0,-3)$ , $(0,-4)$ , ... belong to the set. Also the set is closed since the minimum distance between any two distinct members of $U$ is at least 1.