Analysis: the closure of a set is closed?

[memolee]

Homework Statement

Prove or disprove the following statement:

The closure of a set S is closed.

Homework Equations

Definition of closure: set T is the closure of set S means that T is the union of S and the set of limit points of S.

Definition of a closed set: set S is closed means that if p is a limit point of S then p is in S.

The Attempt at a Solution

So, the closure of set S-- call it set T-- contains all the elements of S and also all the limit points of S. Then, T must have limit points that is contained in it.

I can't prove that all the limit points of T is in T. I can only prove that all the limit points of S is in T. Help?

 

[radou]

You can make this very simple.

The definition of a closure of a set A: the closure of A is the intersection of all closed sets containing A.

In a topological space, what is the intersection of any number of closed sets?

 

[Dick]

That the limit points of S are in T is directly given by the definition of closure. That's not much of a start to the proof. If you want to do this without radou's alternative definition of 'closure', pick p to be a limit point of T. Can't you show that's also a limit point of S? Stating the definition of 'limit point' might help.