# When is the product of closed sets closed in the product topology?

I have a specific example below and i think my proof is wrong because it seems too simple, it would work for the general case which I doubt is true. I would also be interested in the general answer as well; when is the product of closed sets closed in the product topology?

Show that the set $$\prod_{\alpha\in\mathbb{R}} \mathbb{N}$$ is a closed subset of the topological space $$\prod_{\alpha\in\mathbb{R}} \mathbb{R}$$ (endowed with the product topology)

My attempt: $$\mathbb{N}$$ is a closed subspace of $$\mathbb{R}$$ with the canonical topology. In particular, $$\overline{\mathbb{N}}=\mathbb{N}$$. Also since we are in the product topology, it a known fact that closure of a product is the product of the closures. These 2 facts together mean $$\overline{ \prod_{\alpha\in\mathbb{R}} \mathbb{N} }= \prod_{\alpha\in\mathbb{R}} \overline{\mathbb{N}} = \prod_{\alpha\in\mathbb{R}} \mathbb{N}$$ Since $$\prod_{\alpha\in\mathbb{R}} \mathbb{N}$$ contains its closure, it is closed

If that fact about closures is known to you (and it is indeed true), then this is a valid argument to show $$\prod_{\alpha \in \mathbb{R}} \mathbb{N}$$ is closed in $$\prod_{\alpha \in \mathbb{R}} \mathbb{R}$$.