Exterior and limit points: Quick Question

[Buri]

Is the following true?

X\lim(A) = ext(A), where A is a subset of a metric space X.

I think I've found a proof, but I don't feel very secure about it. So could someone just tell me if this is a correct assertion? What I'm trying to prove is that lim(A) is closed.

Thanks

 

[ystael]

No, this is false. The exterior of A can be expressed as X minus something, but that "something" is not the set of limit points of A. (As a hint, think about isolated points.)

 

[Buri]

Ahh really? Hmm I'm going to ahve to think about it more then...damnit

 

[HallsofIvy]

In R, consider $$A= [0, 1]\cup \{2\}$$.

 

[Buri]

Yup, those damn isolated points. Thanks!