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I am considering the set of nondecreasing functions defined on $[0,1]$. These functions are bounded by known constant $M$. Is this set of functions compact in any metric?

I am interested if there are some well known results in the corresponding literature on this on how to choose a metric (or topology) to obtain compactness of this set. I don't know infinite-dimensional analysis that well, I would not know where to look.

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    What about the topology of pointwise convergence? Isn't the set of increasing functions $[0,1]\to [-M,M]$ closed in $\mathbb R^{[0,1]}$? Then it will be compact by Tychonov. – 2017-02-07
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    @jochen the topological space is compact. But, is it metrizable? – 2017-02-07
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    No, it's not metrizable. – 2017-02-07
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    Any set whose cardinal is that of $\mathbb R$, such as your set ot functions, can be a compact metric space under some metric. But I expect you want a metric that has some algebraic or analytic relationship with the the functions. – 2017-02-07
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    @user254665 Yes, it wold be nice to have a metric that is easily interpretable. – 2017-02-08

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