The structure of a first-countable product space. If a product space is first-countable, how to prove all but countably many of the factors have the trivial topology?
 A: HINT: You should try to prove the contrapositive. Let $\{\langle X_i,\tau_i\rangle:i\in I\}$ be a family of topological spaces, let $X=\prod_{i\in I}X_i$ with the product topology $\tau$, and let $$I_0=\{i\in I:\tau_i\text{ is not the trivial topology}\}\;.$$ Show that if $I_0$ is uncountable, then $X$ cannot be first countable. I’ll point you in the right direction.


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*For each $i\in I_0$ choose a point $p_i\in X_i$ that has an open nbhd $V_i\ne X_i$, and for each $i\in I\setminus I_0$ let $p_i$ be any point of $X_i$. Let $p=\langle p_i:i\in I\rangle\in X$. Let $\{B_n:n\in\Bbb N\}$ be any countable family of open nbhds of $p$.

*Show that for each $n\in\Bbb N$ there are a finite $F_n\subseteq I_0$ and open sets $U_i^{(n)}\in\tau_i$ for each $i\in F_n$ such that if $U_i^{(n)}=X_i$ for each $i\in I\setminus F_n$, then the basic open set $U_n=\prod_{i\in I}U_i^{(n)}$ is an open nbhd of $p$ contained in $B_n$.

*Explain why $I_0\setminus\bigcup_{n\in\Bbb N}F_n\ne\varnothing$, and fix $i_0\in I_0\setminus\bigcup_{n\in\Bbb N}F_n$.

*Use $V_{i_0}$ to build an open nbhd $V$ of $p$ in $X$ that does not contain any of the sets $U_n$. Conclude that $V$ does not contain any of the sets $B_n$, and hence that $\{B_n:n\in\Bbb N\}$ is not a local base at $p$.
