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Math Help - R^w in the product topology?

  1. #1
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    R^w in the product topology?

    Prove that R^w (that is, product of Real numbers) in the product topology is path connected.
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    Senior Member Tinyboss's Avatar
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    What have you tried? Have you thought of any ways to use the fact that R is path connected?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by iamthemanyes View Post
    Prove that R^w (that is, product of Real numbers) in the product topology is path connected.
    Not to step on Tinyboss's toes, but perhaps a small hint. I general, if you have a family of (continuous) maps \left\{f_\alpha:X\to X_\alpha\right\}_{\alpha\in\mathcal{A}} then there of course is a natural map \displaystyle F:X\to\prod_{\alpha\in\mathacal{A}}X_\alpha,\; F(x)= (f_\alpha(x)) and if we give \displaystyle \prod_{\alpha\in\mathcal{A}}X_\alpha then we know that F is continuous since \pi_\alpha\circ F=f_\alpha. What do you think?
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