When formally defining Betti number, we often use homology group - but I am not sure how we can use that definition to prove the informal definition of Betti number - that talks about "unconnected and connected" ones.
from Wikipedia (http://en.wikipedia.org/wiki/Betti_number )
Informally, the $k$th Betti number refers to the number of unconnected $k$-dimennsional surfaces.
Actually, I also feel confused about what this actually means. (Example of torus would be appreciated for an explanation. I am having a hard time understanding what "one-dimensional" holes of a torus exactly refers to, and same with two-dimensional holes.)
addition: But what exactly is an outside hole of a torus? I only see one hole inside.... (I know I am stupid for not figuring out, but I just cannot find it...)