I would like to know if there is any reference to the proof of this proposition
"In $\mathcal{H}([0,1])$ (the group of homeomorphisms of the unit interval [0,1]) the product topology and the uniform topology are the same."
or to any general result that contains this case. Thank u.
The straightforward proof is rather simple. Indeed, let $X=\mathcal{H}([0,1])$. Since the uniform topology on $X$ is stronger than the product topology, it suffices to prove that the product topology on $X$ is stronger than the uniform topology. Let $f\in X$ be any function and $U$ be any neighborhood of $f$ in the uniform topology. There exists $\varepsilon>0$ such that $U_\varepsilon\subset U$, where $$U_\varepsilon=\{g\in X: |g(x)-f(x)|<\varepsilon\mbox{ for each } x\in [0,1] \}.$$ Since the function $f$ is continuous on the compact space $[0,1]$, it is uniformly continuous, so there exists natural $n$ such that $|f(x)-f(y)|<\varepsilon/2$ for each $x,y\in [0,1]$ with $|x-y|<\tfrac 1n$. Put $$V=\{g\in X: |g(i/n)-f(i/n)|<\varepsilon/2\mbox{ for each nonnegative integer }i\le n\}.$$ Then $V$ is a neighborhood of $f$ in the product topology on $X$. We claim that $V\subset U$. Indeed, let $g\in V$ be any function and $x\in [0,1]$ be any number. Pick a nonnegative integer $i\le n-1$ such that $x\in [i/n,(i+1)/n]$. Since both $f$ and $g$ are monotonic, $f(x)$ is between $f(i/n)$ and $f((i+1)/n)$ and $g(x)$ is between $g(i/n)$ and $g((i+1)/n)$. Thus there exist $\lambda,\mu\in [0,1]$ such that $f(x)=\lambda f(i/n)+(1-\lambda)f((i+1)/n)$ and $g(x)=\mu g(i/n)+(1-\mu)g((i+1)/n)$. Then $$|f(x)-g(x)|=|\lambda f(i/n)+(1-\lambda)f((i+1)/n)-(\mu g(i/n)+(1-\mu)g((i+1)/n))|=$$ $$|(\lambda-\mu) f(i/n)+\mu(f(i/n)-g(i/n))+ (1-\mu)(f((i+1)/n))-g((i+1)/n))+(\mu-\lambda) (f((i+1)/n))|=$$ $$|(\lambda-\mu) (f(i/n)- f((i+1)/n))+\mu(f(i/n)-g(i/n))+ (1-\mu)(f((i+1)/n))-g((i+1)/n)) |\le $$ $$|(\lambda-\mu)(f(i/n)- f((i+1)/n))|+|\mu(f(i/n)-g(i/n))|+ |(1-\mu)(f((i+1)/n))-g((i+1)/n))|\le $$ $$|\lambda-\mu||f(i/n)- f((i+1)/n)|+ \mu|f(i/n)-g(i/n)|+ (1-\mu)|f((i+1)/n))-g((i+1)/n)|\le $$ $$|\lambda-\mu|\varepsilon/2 + \mu\varepsilon/2+ (1-\mu) \varepsilon/2\le \varepsilon.$$