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.
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
Some problems about path connected and projection maps
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general-topology