R^w in the product topology?

**iamthemanyes** · May 1st 2011, 05:31 PM

Prove that R^w (that is, product of Real numbers) in the product topology is path connected.

**Tinyboss** · May 1st 2011, 06:44 PM

What have you tried? Have you thought of any ways to use the fact that R is path connected?

**Drexel28** · May 2nd 2011, 03:11 PM

Originally Posted by iamthemanyes

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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