The standard definition of a closed manifold is the following:
A closed manifold is a manifold without boundary that is compact.
I am wondering, why we request that the manifold is compact. As I understand, we try to generalise the idea of closed curves, such as circle, and closed surfaces, such as torus. (In this context, I think of closed (for curves) as if we are at one point on a curve and move in either direction for long enough, we will once come back to this same point). Now, it's obvious to request that the manifold has no boundary. This is also clearly not enough, as we have simple objects, such as open unit balls, which have no boundaries, yet they are not "closed", as we intuitively imagine.
So I guess because of this we require that the manifold is compact. But why compact? Why not closed? Or some other condition. What makes compact the right choice? I would appreciate if someone can motivate this definition and (at least intuitively) explain why compact property gives us exactly what we wish.