I was asked a question by someone the other day regarding the topology of pointwise convergence, and I can't seem to get anywhere with it. I was wondering if anyone could be of any assistance...
The question was: is $C[0,1]$, the set of continuous real-valued functions from $[0,1]$ to $\mathbb{R}$, with the topology of pointwise convergence (defined by the sub-basis $A_{a,x,b} :=\{f\in C[0,1] : a< f(x) < b\}$ for $x \in [0,1]$ and $a) a normal topological space?
Wikipedia suggests that $C(\mathbb{R})$ is not normal, so I've tried to show this as a starting point, and then work out whether the same argument holds or fails when we restrict to $[0,1]$: but I can't really get anywhere with this either.
My gut instinct seems to tell me that $C[0,1]$ is not normal, and I have some rather vague hand-wavey intuition for why but I can't really pin anything down at all and have given myself a bit of a headache.
Could anyone point me in the right direction with a good hint or reference please? The question is quite interesting so I'd prefer to try and finish it myself than be given a full solution.
Thanks!