The representation of closed set We already know that closed set is the complement of open set.
But I want to know the representation of closed set without the fact above in the usual topology.
For instance, the integers $\mathbb{Z}$ is closed due to the fact that $\mathbb{Z}$ can be represented by the union of one point sets $\{n\}$
(I don't want to use the fact that "the union of open intervals $(n, n+1)$ is open").

Here is one representation: A closed set is the set with all its accumulations. 
Note that $Z$ is closed not because that it is the union of one point set. If a set is a finite union of one point set in the $T_1$ space, then it must be closed. However, if it is a countble union of one point set, it need not be closed.

Example: $\mathbb R$ with usual topology, the subset $\mathbb Q$ is the union of one point set, however it is not closed.

