Suppose we have $X_1, \dots , X_n$ be locally compact topological spaces. We can Show that $X_1 \times \dots \times X_n$ is locally compact. By taking product of respective Compact sets which containing the ngbhd of the point.
I was wondering what will happen if I take arbitrary product. It feels like same proof works.
No, it does not work, because, in general, a product of neighborhoods is not a neighborhood (because, in general, a product of open sets is not an open set).
Actually, given a family of locally compact spaces, their Cartesian product is locally compact (with respect to the product topology) if and only if only finitely many of them are not compact.