I'm perhaps losing myself in a glass of dirty water but I have a weird doubt. Is the set $(0, 1) \cup \mathbb{N}$ closed?
I mean if I had $[0, 1] \cup \mathbb{N}$, then I would know that it's closed since it's a finite union of closed sets.
On the other side, $\partial (0, 1) = \{0; 1\} \subset (0, 1) \cup \mathbb{N}$ (my definition of the Natural set contains the zero). So the set is closed because it does contain its boundary. Yet $(0, 1) \cup \mathbb{N}$ is not a union of closed sets.
On the other side again, I think of $[0, 1) \cup [1, 3] = [0, 3]$ which is indeed closed...
It puzzles me. But anyway, am I failing somewhere?