I have just realized that the group $\mathsf{musco}(X)$ carries a natural supremum metric $$\rho(f,g)=\sup_{x\in X}\{|y|:\exists x\in X\mbox{ with }y\in (f-g)(x)\}.$$

I have just realized that the group $\mathsf{musco}(X)$ carries a natural supremum metric $$\rho(f,g)=\sup_{x\in X}\{|y|:\exists x\in X\mbox{ with }y\in (f-g)(x)\}=\sup\{|f(x)-g(x)|:x\in\mathsf{single}(f)\cap\mathsf{single}(g)\}.$$