if (X1 x X2 x...x Xn, T) is connected. then each (Xi, Ti) is connected. Let (Xi, Ti) be a topological space. Prove that if (X1 x X2 x...x Xn, T) is connected. then each (Xi, Ti) is connected.
Proof: Since (X1 x X2 x...x Xn, T) is connected, the space has only two clopen sets: the empty set and X1 x X2 x...x Xn iff each Xi is a clopen set or the empty set,i=1, 2, ..,n
Therefore each (Xi, Ti) is connected.
Could you please check this for me? I'm not sure about my proof. If it's not ok, could you please give me some ideas?
Thanks a lot
 A: I’m afraid that it’s not okay: you really haven’t said anything that justifies the conclusion that $X_1$, say, is connected, because you haven’t shown how connectedness of the product is related to connectedness of the factors. How does the fact that $X_1\times\ldots\times X_n$ is the only non-empty clopen set in the product actually imply that $X_1$ is the only non-empty clopen set in $X_1$, for instance?
HINT: Suppose that $H$ is a non-empty clopen subset of $X_k$ for some $k\in\{1,\ldots,n\}$. What can you say about 
$$\pi_k^{-1}[H]=\{\langle x_1,\ldots,x_n\rangle\in X_1\times\ldots\times X_n:x_k\in H\}\;,$$
where $\pi_k:X_1\times\ldots\times X_n\to X_k$ is the usual projection map?
A: I don't really understand your proof as it's currently written.

One different approach is this: Suppose that $X_1$ were disconnected; then there exist open sets $U$ and $V$ such that $U \cap V = \emptyset$ and $U \cup V = X_1$. Then what can you say about the sets
$$U \times X_2 \times \dots \times X_n$$
and $$V \times X_2 \times \dots \times X_n ?$$
A: Yet another approach is that the projection map $\pi_i$ onto the $i-$th coordinate is a continuous map, so that it takes connected sets to connected sets. Then $$\pi_i (X_1 \times X_2 \times.....\times X_n)$$ =$X_i$ is connected, as the continuous image of the connected space $X_1 \times X_2 \times...\times X_n$ 
