Regarding the definition of totally bounded spaces

The definition of totally bounded subset $A$ of a metric space according to most online sources was that for any given $e>0$ there exists a finite number of spheres of radius $e$ with centers in the metric space such that their union contains $A$. However my professor insisted that the centers of the spheres must belong to the $A$. I request anyone to shed some light on this matter.( by a sphere I mean the usual open balls)

• I assume you mean "ball" rather than "sphere" since one doesn't generally use spheres in definitions and theorems about metric spaces. – C Monsour Apr 20 '18 at 3:07
• Contrary to what I previously thought, it appears that your professor is correct. The centers of the open balls must be elements of $A$. Here's a reference (see page $21$):$\;$eml.berkeley.edu/~cshannon/e204_11/lec6sl.pdf – quasi Apr 20 '18 at 3:49

I think your professor is right in the sense that you don't want whether $A$ is totally bounded to depend on the space $A$ is in, so his is a better way of phrasing the definition. However, the definitions are equivalent. Obviously, if $A$ is totally bounded in your professor's definition, it is totally bounded in the web's definition. On the other hand, if it is totally bounded in the web's definition, consider any $\epsilon>0$. $A$ is covered by a finite number of $\epsilon/2$ balls in the ambient space. You can discard any of those balls that don't meet $A$. From each of the remaining ones, select a point in the ball that is also in $A$. Then that finite set of points is a set of centers in $A$ of $\epsilon$ balls that cover $A$ (since an $\epsilon$ ball centered at $x$ contains any $\epsilon/2$ ball that contains $x$).
Engelking ,General Topology, p266 defines it as "a metric space $(X,\rho)$ is uniformly bounded iff for every $\varepsilon >0$ there is a finite $A \subseteq X$ that is $\varepsilon$-dense". The latter means that for every $x \in X$ there exists some $a \in A$ such that $\rho(x,a) < \varepsilon$. (This essentially states that $\{B(a, \varepsilon): a \in A\}$ covers $X$).
So it's defined as a property of a metric space, and when we have a subset $A$ of a metric space, we restrict the metric on $X$ to $A$ and consider that set with the restricted metric as a metric space in its own right. If we then see what $A$ is totally bounded" means, we exactly get the "finite cover of balls with centres in $A$" definition. As Monsour noted, this is equivalent to the one with any open balls around any point of a larger space, but the generally used definiion (as Engelking uses it) is more "intrinsic", as there is no separate notion of a totoaly bounded subset of $X$, just of a totally bounded metric space.