Is topology of pointwise convergence metrizable on the set of functions $f: [0,1] \to [0,1]$? Let $X$ be the set of all monotone functions from $[0,1]$ to $[0,1]$. We know from Helly's selection theorem that $X$ is compact in the topology of pointwise convergence. Now I have the following two questions:


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*Is this topology metrizable? If yes with which metric?

*Is monotonicity needed for $X$ to be compact? Could one apply Tychonoffs theorem to show that all functions from $[0,1]$ to $[0,1]$ are compact?

 A: 
  
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*Is this topology metrizable? If yes with which metric?
  

The space you refer to is also know as the Helly space and according to L. Drewnowski, A. Michalak "Generalized Helly spaces, continuity of monotone functions, and metrizing maps", Fundamenta Mathematicae 200 (2008) the space is "separable, nonmetrizable, first-countable compact Hausdorff". The paper claims that the proof can be found in J. L. Kelley "General Topology" (which I haven't checked).


  
*Is monotonicity needed for $X$ to be compact? Could one apply Tychonoffs theorem to show that all functions from $[0,1]$ to $[0,1]$ are compact?
  

Yes, Tychonoff's theorem does imply that the space is compact. But, as mentioned by @KaviRamaMurthy in comments, Helly's theorem talks about sequential compactness. Note that in general, nonmetrizable case neither implies the other.
So is the space of all functions sequentially compact? It seems it is not, since it is listed in the famous "Counterexamples in Topology" book as a compact but not sequentially compact space.
