Proposition 10

In a product space $X = \prod_{i \in J} X_i$, the component of $x = (x_i)$ in $X$ is the product of the components of $x_i$ in the factors $X_i$.

This product set is connected (no. 4, Proposition 8). Conversely, if $A$ is a connected subset of $X$ which contains $x$, then $\operatorname{pr}_i(A)$ is a connected set (no. 2, Proposition 4) which contains $x$; since $A \subset \prod_i \operatorname{pr}_i(A)$, it follows that $A$ is contained in the product of the components of the $x_i$.