Whilst seeking to recall a few things about metric spaces, I ran into the embarrassing problem of not being able to grasp the notion behind a certain problem.

I’m having a hard time trying to find an infinite collection of open sets whose intersection is not open, when we’re considering the set $\mathbb{R^2}$ with the discrete metric$$d(x,y)=\begin{cases}
0 & \text{ if } x=y, \
1 & \text{ if } x\neq y.
\end{cases}$$What am I missing?

Question

Whilst seeking to recall a few things about metric spaces, I ran into the embarrassing problem of not being able to grasp the notion behind a certain problem.

I’m having a hard time trying to find an infinite collection of open sets whose intersection is not open, when we’re considering the set $\mathbb{R^2}$ with the discrete metric$$d(x,y)=\begin{cases}
0 & \text{ if } x=y, \
1 & \text{ if } x\neq y.
\end{cases}$$What am I missing?

across · Answer

Thus far, what I’ve managed to gather is that it may be possible for every subset of $\mathbb{R^2}$ with this metric to be an open set, but I’m finding it difficult to formalize that.

For example, if I let any point $p\in\mathbb{R^2},$ I’ll definitely see that $N(p;1)=\{p\}$.

across · Answer

However, were this to be true, it would imply that there exists no such collection.

I have a headache.

across · Answer

Never mind; I saw it: apparently, there’s indeed no such collection for the very reason I stated in the first comment.

I can be fairly dumb at times!