An infinite union of closed sets is a closed set?

Every subset of $\mathbb R^ n$ is a union of closed sets, namely, the one-point sets consisting of each one of its points.

Yet not all subsets of $\mathbb R^n$ are closed!

Think of the union of the $B_{n} = [1/n, 1]$.

Consider the topological space (subspace of$\mathbb{R}$ (real numbers) with the usual topology given by $\epsilon$-neighborhood) given by $$\{0\}\cup\{\frac{1}{n}:n\in\mathbb{N}_{>0}\}$$ Then, for every $n\in\mathbb{N}_{>0}$ the subset $$\{\frac{1}{n}\}$$is both open and closed, but the countable union $$\displaystyle\bigcup_{n>0}\{\frac{1}{n}\}$$ is precisely the set $$\{\frac{1}{n}:n>0\}$$ which is not closed, for example because $0$ lies in its closure, but not in the set itself.