Neighborhoods in the product topology. In the last two lines thread, it is said that $N \times M \subseteq A\times B$ is a neighborhood of $0$ if and only if $N, M$ are neighborhoods of $0$. Here $A, B$ are topological abelian groups. How to prove this result? I searched on the Internet, but was not able to find a proof.

It is important to note that in that thread it is explicitly said that $\,G\times G\,$ has the product topology, meaning that a subset here is open iff it is the union of sets of the form $\;N\times M\;$ , with $\;N,M\subset G\;$ open...This, I think, solves the problem, as it is exactly the same with $\,A\times B\;$ a product of top. groups  :)
