A few questions on the properties of $\mathbb{R} ^ {[0,1]}$ 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. 


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*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!
 A: $X$ is not metrizable since it is not first-countable.
Since $\Bbb R$ is Hausdorff, so is an arbitrary product of $\Bbb R$. More generally, an arbitrary product of Hausdorff spaces is Hausdorff.
If $X$ were compact, another space $Y$ which you could write as the continuous image of $X$ would have to be compact too. Can you guess which map $X\to Y$ I'm talking about?
Local compactness is disproved similarly. Can you think of a reason why no neighborhood in $X$ can be compact?
$X$ is path-connected. Given points $x=(x_i)_{i\in[0,1]}$ and $y=(y_i)_i$ in $X$, a path can be expressed using the individual paths $x_i\to y_i$ over all $i\in[0,1]$.
Connectedness is a bit more difficult. The usual proof I know of uses the dense connected subset $D=\{(x_i)_i\mid x_i=p_i\text{ for almost all $i$}\}$ where $p=(p_i)_i$ is some fixed point in $X$.
Edit: Najib reminds me that path-connectedness implies connectedness, so the last paragraph applies if you wanted to show connectedness directly, or for the general case of a product of connected spaces.
In the same spirit, if you have disproved local compactness, then you've disproved compactness, since a compact Hausdorff space is always locally compact.
A: 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.
