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  1. Why if $\{A_\alpha\}_{\alpha \in \Omega}$ is a collection of path connected spaces. We must show that $\cup_{\alpha \in \Omega}$ is path connected.

    • I think $\cap_{\alpha \in \Omega}A_\alpha$ must be nonempty, but I can't prove $\cup_{\alpha \in \Omega}$ is path connected.
  2. Prove that $P_\beta : \prod_\alpha X_\alpha \to X_\beta$ is continuous, open and onto for all $\beta.$

    • I can prove $P_\beta$ is continuous and open. But I can't proof $P_\beta$ is onto [I think it's easy.] Please hint me to get $P_\beta$ is onto

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  1. Let $z \in \bigcap_{\alpha\in\Omega} A_\alpha$ and let $x,y \in \bigcup_{\alpha\in\Omega} A_\alpha$. Why is there a path $\gamma_1$ from $x$ to $z$? Why is there a path $\gamma_2$ from $y$ to $z$? Can you make a new path $\gamma$ from $x$ to $y$ that combines $\gamma_1$ and $\gamma_2$ in some way?
  2. What is the definition of $P_\beta$? If I have some $x_\beta \in X_\beta$ and I choose arbitrary points $x_\alpha \in X_\alpha$ for $\alpha \ne \beta$, what is $P_\beta((x_\alpha)_\alpha)$?
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    @TeeMth: If you find this answer helpful, please accept it. You should create a new question about first countable spaces.2012-11-20