Thread: Sets of isolated points

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

My Answer: Yes

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.

Originally Posted by arsenicbear

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

My Answer: Yes

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 has an open neighborhood which only contains THAT POINT (i.e., , it follows that any point has a neighborhood is open .

Tonio