Why to see that $\overline{B}(x;r)$ is closed if it was just defined?

Question

I'm reading Conway's A Course in Point Set Topology. He defines open and closed balls and then he introduces some examples, one of these examples is this:

(c) For any $r>0$, $\overline{B}(x;r)$ is closed. To see this, set $G=X \setminus B(x;r)$ and let $y\in G$; thus, $d(y,x)>r$; Let $0<s<d(x,y)-r$. If $d(z,y)<s$ then $r<d(x,y)-s<d(x,y)-d(y,z)\leq[d(x,z)+d(z,y)]-d(y,z)=d(x,z)$; that is, $B(y;s)\subseteq G$. Since this shows that $G$ is open, it follows that $\overline{B}(x;y)$ is closed.

I'm having a first exposure to it, in the book, he just defined closed balls and then he suggests that one needs to see it. Why is that?

Answer 1

A closed set is a set whose complement is open. He defines a closed ball as a particular set. Calling it a closed ball does not entail that it is a closed set. One first needs to verify that the complement of the closed ball is an open set. If this is confusing, call $\overline {B}(x;r)$ a banana. Now, is the banana closed? well, you need to check something, namely that the complement of the banana is open. That is what he is doing. After you verified the banana is closed, you can remember the banana is actually much better be called a closed ball. And now you can conclude that the closed ball is a closed set.

Answer 2

I would assume that by $\overline{B}(x,r)$ he meas the closure of the open ball, and he wishes to justify that this is indeed closed. (And is a closed ball)

Answer 3

I think he defined open ball as $B$ and then he proved, that a closed ball $\overline{B}$ is a closure of an open ball $B$.

Answer 4

He defined an object called a closed ball; this is the name that he (and every other author) chose to give to the object $\overline{B}(x, r)$ which is a subset in a metric space. He could have given such an object a different name, say a ball of type II. As $\overline{B}(x, r)$ is a subset of a metric space (no matter what you call it), it is either closed or it is not. He then shows that this subset is closed. In hindsight, instead of calling such objects balls of type II, it makes sense to call them closed balls.