Definition
The component $C_x$ of $x\in X$ is the biggest connected subspace of $X$ that contains $x$.
Definition
The quasi-component $Q_x$ of $x\in X$ is the intersection of the clopen sets of $X$ that contains $x$.
Statement
If $\mathfrak{X}=\{(X_j,\mathcal{T}_j): j\in\ J\}$ is a collection of topological spaces then the component $C_x$ and the quasi-componet $Q_x$ of any $x\in X:=\prod_{j\in\ J}X_j$ is equal to the product of the components $C_{x_j}$ and quasi-components $Q_{x_j}$ of the projections $x_j$ of $x$ in $X_j$.
Proof. Cleary $K_x=\prod_{j\in J}C_{x_j}$ is connected in $X$ and so $K_x\subseteq C_x$. Then clearly $\pi_j(C_x)$ is connected in $X_j$ for any $j\in\ J$, since the projections are continuous functions; and so $\pi_j(C_x)\subseteq C_{x_j}$ for any $j\in J$ and so $C_x\subseteq\pi_j^{-1}(C_{x_j})$ for any $j\in J$, that is $C_x\subseteq\bigcap_{j\in J}\pi_j^{-1}(C_{x_j})=K_x$.
First of all I think that my proof is partially uncorrect, since for sake of completeness I should to prove that $K_x$ is embeddable in $X$ and then that $\bigcap_{j\in J}\pi_j^{-1}(C_{x_j})$ is actually equal to $K_x$. Anyway to prove that $K_x$ is embeddable in $X$ I tried to define the funcion $\phi:K_x\rightarrow X$ through the condiction $[\phi(x)](j)=x(j)$ for any $x\in K_x$ for which is obvious that $\phi$ is injective and continuous, since $\pi_j\circ\phi=\pi'_j$ for any $j\in J$, where $\pi_j$ and $\pi'_j$ are the projections in $X$ and $K_x$; then I should to prove that $\phi[K_x]= H_x:=\{z\in X:z(j)\in C_{x_j},\forall j\in J\}$ and so define an inverse of $\phi$. Then as you can see the proof is incomplete, since I didn't prove the statement about the quasi-components: so I ask to complete the proof. So could someone help me, please?
