Why can projection function on $X \times S$ be regarded as a local homeomorphism?

I am studying some properties of local homeomorphism

I am in particular trying to find a local homeomorphism that is not a homeomorphism and the projection function seems to be the perfect candidate since it doesn't "recover".

I found an example here:

For $X$ any topological space and for $S$ any set regarded as a discrete space, the projection

$\pi_X: X\times S \to X$ is a local homeomorphism.

I am trying to see how this function fits the definition:

Definition: Let $(X, \mathcal{T})$ and $(Y, \mathcal{J})$ be topological spaces. A function ${\displaystyle f:X\to Y\,}$ is a local homeomorphism if for every point $x \in X$ there exists an open set $U \subseteq X$ containing $x$ and an open set $V \subseteq Y$ such that the restriction ${\displaystyle f|_{U}:U\to V\,}$ is a homeomorphism.

So let $V_x$ be an open set on $X$, and $U_x$ some open set in $X \times S$ containing $x$, we wish to show that $\pi_X|_{U_x}: U_x \to V_x$ is a homeomorphism

1. Show $\pi_X|_{U_x}$ is continuous: Take an open set in $V_x$, then such set has the form $V_x \cap W$, $W$ is open in $X$, $\pi_X|_{U_x}^{-1}(V_x \cap W) = \pi_X|_{U_x}^{-1}(V_x) \cap \pi_X|_{U_x}^{-1}(W) = \pi_X|^{-1}(V_x) \cap U_x \cap \pi_X|^{-1}(W)$, right handside is open in the product topology on $X \times S$