TITLE: Closed property of nonempty finite set
QUESTION [7 upvotes]: I came across this text in Rudin's book where it has been mentioned that a non-empty finite set is closed. 
But a closed set is a set which contains all of it's limit points in the set itself but none of the elements of a non-empty finite set can possibly have a limit point because a neighborhood of a limit point has infinite points . So how come a non-empty finite set be a closed set ??

REPLY [4 votes]: Alternatively a set $A$ is closed iff $A=\bar A= A \cup \{\text{set of all limit points}\}.$
In your case, $\{\text{set of all limit points}\}=\emptyset.$ Thus $A=\bar A$.