Anyway, thanks very much for your detailed response.
The fact that there are so many equivalent definitions kind of bothers me. Its like everything becomes kind of circular. For example when I was learning Analysis(I am still learning it, in fact), one book would define a closed set as the complement of an open set and it would ask me to prove that a closed set contains all of its limit points. Another book would instead define a closed set as a set which contains all of its limit points and ask me to prove that its complement is open. I was like, "What's the point to all this?". I had fun proving it, but it seemed pointless.