# Thread: Sets of isolated points

1. ## Sets of isolated points

If we have a metric space X, is S, the set of isolated points of X, open?

My Logic: Every point of S has a neighborhood, which is empty. The empty set is a subset of every set. Therefore every point of S has a neighborhood that is a set of S.

I just want to make sure I did not misunderstand anything.

2. Originally Posted by arsenicbear
If we have a metric space X, is S, the set of isolated points of X, open?

My Logic: Every point of S has a neighborhood, which is empty. The empty set is a subset of every set. Therefore every point of S has a neighborhood that is a set of S.

I just want to make sure I did not misunderstand anything.

Almost, but not quite imo: since any point $\displaystyle s\in S$ has an open neighborhood $\displaystyle U_s$ which only contains THAT POINT $\displaystyle s$ (i.e., $\displaystyle U_s = U_s\cap X=\{s\}$, it follows that any point $\displaystyle s\in S$ has a neighborhood $\displaystyle U_s=\{s\}\subset S \Longrightarrow S$ is open .

Tonio