TITLE: Is a set closed if it has no accumulation points?
QUESTION [5 upvotes]: I was wondering 

if a set $A$ has no accumulation point, is this set $A$ closed?

I think this is true, but I'm not quite sure.
Here's my thinking:
By closed set definition: A set $A$ is closed if every accumulation point of $A$ is a point of $A$. 
Since $A$ has no accumulation points, it is closed. Am I saying this right?

REPLY [3 votes]: The set of all accumulation points of S is the empty set and every set has the empty set as a subset. Therefore, S contains all its accumulation pointz and thus S is closed. Very simple.