$\mathbb{R^w}$ is connected in product topology but what about the path components of $\mathbb{R^w}$ in product topology?
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In a product of path-connected spaces, what would be the natural candidate for a path connecting two points? Would that work? If not, how would it fail? – Daniel Fischer Nov 14 '17 at 16:30
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No that may not work...For finite product of path connected set it would be path connected but is it true for infinite product? – Samiron Parui Nov 14 '17 at 16:50
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Yes it also works for infinite product because infinite product of continuous map is also continuous. i.e a mapping from $\mathbb{R}$ to $\mathbb{R^w}$ with all its components continuous is continuous. I am sorry .I should think before posting this question... – Samiron Parui Nov 14 '17 at 17:01