Remark 3

Let $(X_j)_{j\in J}$ be a family of topological spaces. Let X denote the product topological space $\prod_{j\in J}X_j$ and let $x= (x_j)$ be a point of X.

Endow the sets $\pi_1(X, x)$ and $\pi_1(X_j, x_j)$ with the quotient topology of the topology of compact convergence on the spaces $\Lambda_x(X)$ and $\Lambda_{x_j}(X_j)$. The isomorphism $\pi_1(X, x)\rightarrow \prod_{j\in J}\pi_1(X_j, x_j)$ is then a homeomorphism. It is continuous (III, p. 294, remark 1). The topology of compact convergence on Λ(X) is generated by the subsets of the form $\mathbf{T}(K,U)$, where K is a compact subset of $\mathbf{I}$ and U an open subset of X. For $j\in J$, let $U_j$ be an open subset of $X_j$, such that $\prod_{j\in J}U_j\subset U$. Then (pr$_j$)$_*(\mathbf{T}(K,U))$ contains $\mathbf{T}(K,U_j)$. This shows that the mappings (pr$_j$)$_*: \Lambda_x(X)\rightarrow \Lambda_{x_j}(X_j)$ are open, and the mappings $\pi_1$(pr$_j, x_j$) are also open. Since they are surjective, the mapping $\pi_1(X, x)\rightarrow \prod_{j\in J}\pi_1(X_j, x_j)$ is open (TG, I, p. 34, Prop. 8). Being continuous and bijective, it is a homeomorphism (TG, I, p. 30, example 2).