Proposition 9

The product of a family of locally path-connected spaces, connected with the exception of a finite number of them, is locally path-connected.

Let $(X_j)_{j\in J}$ be a family of locally path-connected spaces which are connected, with the exception of a finite number of them. Let $x=$ $(x_j)_{j\in J}$ be a point of the product space X = $\prod_{j\in J}X_j$. By assumption, a fundamental system of neighbourhoods of $x$ in X consists of sets of the form $V =\prod_{j\in J}V_j$ where, for every $j\in J$, $V_j$ is a path-connected neighbourhood of $x_j$ in $X_j$ and where $V_j= X_j$ except for a finite set of indices $j\in J$ (TG, I, p. 24). These sets being path-connected (III, p. 260, prop. 6), X is locally path-connected.