I am studying The Arzelà-Ascoli Theorem by pdf , and I don´t understand what is definition of tubes in the sentence below.
To prove the theorem, we will start by simplifying it: we show that the closure of Φ is compact if Φ itself is totally bounded. The proof of this lemma makes use of the fact that each function in the closure set is a limit point of Φ and thus close to a function in Φ. Hence, if Φ is covered by a finite number of ε-tubes, the closure set is covered by ever so slightly wider tubes, centered about the same functions. In addition, the closure set is complete, being a closed subset of a complete space. The compactness of the closure set will follow immediately, since a complete totally bounded set is compact in any metric space.