$E$ is a compact metric space. Consider a compact set $A \subset C(E)$ where $C(E)$ denotes the set of all continuous functions on $E$. Since $A$ is compact, any open cover of $A$ has a finite subcover.
My question is what does it mean to say that " consider a finite collection of open balls of a specified radius centered at the functions which cover $A$". I cannot understand what it means to say open balls centered at functions, doe sit mean for every function and every point of this function, I need to create an open ball.
All this is in context of a metric given by the sup norm