Show that every open set $A$ is in a metric space $(X,d)$ is the union of closed sets.
This is a question on my analysis homework. I understand that this can only be true if we consider the union of infinite closed sets. However, I am not sure what I can do. I understand to prove a set is open, then a ball centered at an arbitrary point with a radius will be completely contained in the set. But what is the radius? Is this the correct approach?
Best Answer
In fact, every subset of a metric space is the union of closed sets.
Hint: In a metric space singleton sets $\{x\}$ are closed.