Lemma 3

Let $(U_i)_{i\in I}$ be a finite family of nonempty open subsets of a topological space X and let O be an open dense subset of $X\times X$. There exists a family $(V_i)_{i\in I}$ of nonempty open subsets of X such that $V_i\subset U_i$ for all $i\in I$ and such that $V_i\times V_{i'}\subset O$ for every pair $(i, i')$ of distinct elements of I.

For every pair $(j_1, j_2)$ of distinct elements of I, the set of families $(x_i)\in X^I$ such that $(x_{j_1}, x_{j_2})\in O$ is an open dense subset of $X^I$. The intersection Ω of these open sets, when $(j_1, j_2)$ runs through the finite set of pairs of distinct elements of I, is therefore an open dense subset of $X^I$. Consequently, $\Omega \cap \prod_{i\in I}U_i$ is a nonempty open subset of $X^I$, hence contains an open subset of $X^I$ of the form $\prod_{i\in I}V_i$, where for every $i\in I$, $V_i$ is an open, nonempty, subset of $U_i$. This proves the lemma.