Proposition 8

Every product of connected spaces is connected. Conversely, if a product of non-empty spaces is connected, then each of the factors is connected.

Let $X = \prod_{i \in I} X_i$ be a product of topological spaces. If the $X_i$ are non-empty, we have $X_i = \operatorname{pr}_i X$ for each $i \in I$; hence if $X$ is connected so are the $X_i$ (no. 2, Proposition 4). Conversely, suppose that each of the $X_i$ is connected and $X$ is not. By Proposition 5 of no. 2, there exists a continuous surjective mapping $f : X \to X'$, where $X'$ is a discrete space which contains more than one point. Let $a = (a_i)$ be any point of $X$, and $x$ any index; the partial mapping $f_x : X_x \to X'$, defined by $f_x(x) = f((y_i))$ where $y_x = x$ and $y_i = a_i$ if $i \neq x$, is continuous on $X_x$; since $X_x$ is connected, $f_x$ must be constant on $X_x$. It follows immediately by induction that $f(x) = f(a)$ for all points $x = (x_i)$ such that $x_i = a_i$ for all but a finite number of indices $i \in I$. But these points $x$ form a dense subset of $X$ (§ 4, no. 3, Proposition 8). Hence $f$ is continuous on $X$ and constant on a dense subset of $X$, and therefore constant on $X$ (§ 8, no. 1, Proposition 2, Corollary 1). But this contradicts the definition of $f$.