In the book Introductory Real Analysis (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"
Expanding on my comment:
For every subset $A$ of a topological space, $\bar{A} = A\cup A'$, where $\bar{A}$ is the closure of $A$ and $A'$ is the set of its limit points.
Now, we suppose that $X_n$ has no limit points, then $\overline{X_n} = X_n$, and then $X_n$ is closed, since the closure of every set is closed.
EDIT: I wrote this assuming the authors of the text meant limit point as in here, which does not seem to be the case, see the comments below this answer.