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 $\displaystyle \left\{f_\alpha:X\to X_\alpha\right\}_{\alpha\in\mathcal{A}}$ then there of course is a natural map $\displaystyle \displaystyle F:X\to\prod_{\alpha\in\mathacal{A}}X_\alpha,\; F(x)= (f_\alpha(x))$ and if we give $\displaystyle \displaystyle \prod_{\alpha\in\mathcal{A}}X_\alpha$ then we know that $\displaystyle F$ is continuous since $\displaystyle \pi_\alpha\circ F=f_\alpha$. What do you think?