Uncountable Product of First-Countable Spaces is not always First-Countable

Theorem
Let $I$ be an indexing set with uncountable cardinality.

Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$.

Let each of $\struct {S_\alpha, \tau_\alpha}$ be a first-countable space.

Then it is not necessarily the case that $\struct {S, \tau}$ is also a first-countable space.

Proof
Let $T = \struct {\Z_{\ge 0}, \tau}$ denote the topological space consisting of the set of positive integers $\Z_{\ge 0}$ under the discrete topology.

Let $I$ be an indexing set with uncountable cardinality.

Let $T' = \struct {\ds \prod_{\alpha \mathop \in \mathbb I} \struct {\Z_{\ge 0}, \tau}_\alpha, \tau'}$ be the uncountable Cartesian product of $\struct {\Z_{\ge 0}, \tau}$ indexed by $I$ with the product topology $\tau'$.

From Discrete Space is First-Countable, $T$ is a first-countable space.

But from Uncountable Cartesian Product of Discrete Topology on Positive Integers is not First-Countable, $T$ is not a first-countable space.

Hence the result.