Definition:
Let $(X_i,\tau_i)_{i\in I}$ be a collection of topological spaces. We will define a topology on $X_I$ in the following manner:
Let $\S$ $=$ $\{$ $\pi_i^{-1}(U_i)$ $:$for all $i\in I$, $U_i$ is open in $X_i$ $\}$
Then the topology generated by the subbasis $\S$ on $X_I$ is called the $\textbf{product topology}$. We will call $X_I$ the $\textbf{product space}$.
Now, we the following is a basis for the topology generated by $\S$:
$\mathbb{B}=$ $\{$ $\bigcap_{i=1}^nS_i$ $:$ $S_i\in \S$ , $n\in \mathbb{N}$ $\}$. Hence, unpacking the definition, we obtain:
$\mathbb{B}$ $=$ $\{$ $\bigcap_{i=1}^n \pi_{m_i}^{-1}(U_{m_i})$ $:$ $U_{m_i}$ is open in $X_{m_i}$ , $n\in \mathbb{N}$ and $m\in I$ $\}$ is a basis for the topology generated by the subbasis.
I would like someone to confirm that what I have written is correct, please.
So, the product topology is the coarsest that makes the projections continuous.