TITLE: A few questions on the properties of $\mathbb{R} ^ {[0,1]}$
QUESTION [5 upvotes]: Given the topological space $X=\mathbb{R}^{[0,1]}$ with the product topology, there are several properties regarding to  $X$ which I am not sure if are true/false. 

Is $X$ metrizable? I'm having trouble on how I can prove/disprove this, and I'm not sure if I should aim to proving this or to find a counterexample. 
Is $X$ normal and/or Hausdorff? I think I can show $X$ is Hausdorff, but not quite sure as to this one or on how to formalize such a proof.
Is $X$ compact and/or locally compact? (And does compactness implies locally compactness or is it the other way around?)
Is $X$ connected and/or path-connected?

I find it especially hard on how to start on proving/disproving each, so even a hint will help!:)
Thanks!

REPLY [2 votes]: The space $\mathbb{R}^{[0,1]}$ is just a product of $|\mathbb{R}|$ many copies of $\mathbb{R}$.
A product space $\prod_i X_i$ is connected iff all $X_i$ are connected. $\mathbb{R}$ is connected so.. The same theorem holds for path-connectedness as well (and is easier to prove even).
A product space $\prod_i X_i$ is compact iff all $X_i$ are compact. So the only question that remains: is $\mathbb{R}$ compact?
A product space $\prod_i X_i$ is Hausdorff iff all $X_i$ are Hausdorff. So is $\mathbb{R}$ Hausdorff?
Try showing that $\mathbb{R}^{[0,1]}$ is not first countable. This should kill illusions of metrisability.
The hardest to see is that $\mathbb{R}^{[0,1]}$ is not normal. For this see this question and its answers.