# Prove that set of all limit points is closed?

• May 6th 2012, 12:08 PM
mariogs379
Prove that set of all limit points is closed?
Hey guys,

If A is a subset of R, let L be the set of all limit points. We want to show that L is closed. Here's how I was thinking about proving it:

To show that L is closed, we can show that the complement of L (call it c(L)) is open. c(L) is open if we can put an epsilon-neighborhood around each of the isolated points of L and always have it included in L. I'm not exactly sure where to go from here.

Any hints would be much appreciated.

Thanks,
Mariogs
• May 6th 2012, 12:29 PM
Plato
Re: Prove that set of all limit points is closed?
Quote:

Originally Posted by mariogs379
If A is a subset of R, let L be the set of all limit points. We want to show that L is closed. Here's how I was thinking about proving it: To show that L is closed

Surely you mean that $L$ is the set of all limit points of $A~?$
That is usually called the derived set of $A$ and is noted by $A'$
Now if $t\notin L$ then $t$ is not a limit point of $A$.
So there is open set $Q_t$ such that $O_t\cap (A\setminus\{t\})=\emptyset$.
Can finish?
• May 6th 2012, 01:26 PM
mariogs379
Re: Prove that set of all limit points is closed?
Yeah, that's what I meant. So Qt must be open and, because it is the complement of L, L must be closed. Yeah?