What does convergence mean in the product topology

Question

In regular single topologies (say $\mathbb{R}$, for example), understanding sequence convergence is straightforward.

Excluding the metric space definition and relying solely on a topological definition, a sequence $\langle a_n \rangle$ converges to a point $b$ if and only if $\langle a_n \rangle$ every open set containing $b$ contains all but a finite number of terms of $\langle a_n \rangle$.

In a single topological space, these sequences are generally easy to inspect, in my opinion, because there is only one parameter to keep track of (because there is only one topological space).

But how do we use this definition of convergence to understand how sequences converge and diverge in product topologies?

For example, in $\mathbb{R}$, the sequence $\langle 0.9, 0.99, 0.999, \dots \rangle$ would converge to $1$. But how can we extrapolate this to understand convergence in, say, $\mathbb{R} \times \mathbb{R} \times \dots \times \mathbb{R}$? Or even an arbitrary topological product space $\mathcal{T}_1 \times \mathcal{T}_2 \times \dots \times \mathcal{T}_n$?

What would happen if say, $\mathcal{T}_1$ converges to a point $t \in \mathcal{T}_1$ but $\mathcal{T}_2$ diverges to infinity? Do all topological spaces $\mathcal{T}_i$ in a product space need to converge individually for the product space to have an understanding of convergence?

Answer 1

The product topology is the 'most efficient' way of making all projections continuous. As a corollary, a sequence in the product topology converges if and only if the projections converge in each factor.