Hello,

I have the following problem:

Prove that the set of limit points E' of a subset E is a closed set.

Help please,

Thank you,

Printable View

- Mar 4th 2010, 01:57 PMmohammadfawazBasic topology
Hello,

I have the following problem:

Prove that the set of limit points E' of a subset E is a closed set.

Help please,

Thank you, - Mar 4th 2010, 02:02 PMmabruka
What definition of closed are you using?

I think the complement of an open set right? - Mar 4th 2010, 02:11 PMPlato
- Mar 4th 2010, 04:29 PMDrexel28
- Mar 4th 2010, 05:40 PMsouthprkfan1
Let (xn) be a convergent sequence in E', (xn) --> x. We want to show x is in E'. I.e, we want to show x is a limit point of E.

We know x is a limit point of E if given any e>0 there is a point in E that is in the ball of radius e around x. That is, there exists a p in E such that:

d(x,p) < e

Heres the proof:

Since (xn) --> x, then there is a k where d(x,xk) < e/2 *

and since xk in in E', it is a limit point of E, so there is a point p in E such that: d(xk,p) < e/2 **

I claim: d(x, p) < e

Pf.

d(x,p) <= d(x,xk) + d(xk, p) < e/2 + e/2 (by * and **)

so d(x, p) < e

Edit: Put Metric d instead of absolute value, if that's not what you mean, just replace every d(x,y) with lx - yl - Mar 4th 2010, 05:41 PMDrexel28