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These sets of finite sequences are referred to as cylinder sets in the product topology. en.wikipedia.org

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. en.wikipedia.org

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). en.wikipedia.org

This topology is called the product topology. en.wikipedia.org

The simplest option is to take the usual product topology. en.wikipedia.org

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). en.wikipedia.org

For example, in finite products, a basis for the product topology consists of all products of open sets. en.wikipedia.org

There is a natural topology on this space, called the product topology. en.wikipedia.org

This stems directly from the fact that, in the product topology, almost all the factors in a basic open set are the whole space. en.wikipedia.org

The subspace topology and product topology constructions are both special cases of initial topologies. en.wikipedia.org

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