In the book [Introductory Real Analysis](http://rads.stackoverflow.com/amzn/click/0486612260) (Kolmogorov & Fomin, English Translation by Richard Silverman Pg 95, see below)
It says:

"the sets $X\_n= \{x\_n, x\_{n+1}, ...\}$ form a centered system of ***closed*** sets." I am not understanding why these countable sets are closed in this space.

Obviously, for any point not in $X\_n$, I can construct an open set $O$ containing only a finite set of points of $X\_n$, and so I can eliminate those points to construct $X\_n$ to have a null intersection with $O$. But I can't do that for *every point* without obliterating the $X\_n$'s.

Can anyone tell me then why these $X\_n$ sets are closed?

FYI They define a limit point of a set as "every neighborhood of the point has infinite elements of the set"

[![Page94-95](https://i.stack.imgur.com/YHxd0.png)](https://i.stack.imgur.com/YHxd0.png)