The hoop picture works for convergence in the product topology as well: here we only require all the functions to jump through any given "finite" set of hoops.
Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given the discrete topology).
Algebraically, this corresponds to the additive group being dense in the profinite integers (direct product of the -adic integers over all primes, with the product topology).