Proposition 13

a) Let $(X_i)_{i \in I}$ be a family of locally connected spaces such that $X_i$ is connected for all but a finite number of indices $i \in I$. Then the product space $X = \prod_{i \in I} X_i$ is locally connected.

b) Conversely, if the product of a family $(X_i)$ of non-empty topological spaces is locally connected, then each $X_i$ is locally connected, and $X_i$ is connected for all but a finite number of indices.

a) Let J be the finite subset of I such that $X_i$ is not connected if and only if $i \in J$. Let

be an elementary set containing a point $x = (x_i)$ of X and let K be the finite subset of I such that $U_i \neq X_i$ if and only if $i \in K$. Let $V_i$ be $X_i$ for $i \notin J \cup K$, and let $V_i$ be a connected neighbourhood of $x_i$ contained in $U_i$ for $i \in J \cup K$; then is connected (by Proposition 8 of no. 4) and is a neighbourhood of $x$ contained in U. Hence X is locally connected.

b) Let $a = (a_i)$ be a point of X and let V be a connected neighbourhood of $a$ in X. Since we have $\mathrm{pr}_i V = X_i$ except for a finite number of indices (§ 4, no. 1) it follows from no. 2, Proposition 4 that the $X_i$ are connected, for all but a finite number of indices. On the other hand, for each $x \in I$, each $a_x \in X_x$ and each neighbourhood $V_x$ of $a_x$ in $X_x$, there is a point $x$ of X such that $\mathrm{pr}_x x = a_x$, and

is a neighbourhood of $x$ in X; V therefore contains a connected neighbourhood W of $x$, whose projection $\mathrm{pr}_x W$ is a connected neighbourhood of $a_x$ contained in $V_x$ (no. 2, Proposition 4 and § 4, no. 2, Proposition 5). Hence each $X_x$ is locally connected.