Sub basis for a product topology

Question

Let $(X,\tau_x)$ and $(Y,\tau_Y)$ be two topological spaces. Consider $U\subseteq X$ and $V\subseteq Y$ and $\pi_X: X\times Y \to X$ and $\pi_Y: X \times Y \to Y$ be the natural projections map. Then is $\mathcal S= \{\pi_X^{-1}(U) ,\pi_Y^{-1}(V)\}$ a sub-basis for the product topology on $X\times Y$ ?

Answer 1

Yes, the definition of the product topology says that

$$\mathcal{S} = \{\pi_Y^{-1}[U]: V \subseteq Y \text{ open }\} \cup \{\pi_X^{-1}[U]: U \subseteq X \text{ open }\}$$

is a subbase for the product topology on $X \times Y$.

For general products $X = \prod_{i \in I} X_i$ the subbase is:

$$\bigcup_{i \in I}\{\pi_i^{-1}[U]: U \subseteq X_i \text{ open }\}$$

where $\pi_i: X \to X_i$ is the projection.