What are some examples in which the introduction of nets helped understand a concrete topological space? I've had the chance to learn about nets, though every statement I was exposed to didn't seem to be useful in practice.
For example, the fact $x \in \bar{A}$ iff there exists some net $(x_\alpha)_{\alpha \in J}$ such that $x_\alpha$ converges to $x$, doesn't seem to be useful in practice because if one can construct such a net, than one can also prove directly that every neighboorhood of $x$ has a non empty intersection with $A$.
Question: Are examples of results about concrete spaces (or otherwise — spaces that have some specific property that is not described by the concept of nets) being obtained using nets, and that could not be obtained using sequences, where nets actually prove to be an efficient tool?

You can prove Tychonoff's theorem using nets (that's how it's done in Folland's Real Analysis, for instance), but that can't be done with sequences, even for concrete spaces.
